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Москва Университетская книга
2010
УДК 539.171 ББК 22.383.5 С86
Строковский Е. А. С86 Лекции по основам кинематики элементарных процессов :
учебное пособие / Е. А. Строковский. — М. : Университетская книга, 2010. — 298 с. : табл., ил.
ISBN 978-5-91304-154-8
УДК 539.171 ББK 22.383.5
Учебное издание Евгений Афанасьевич Строковский Лекции по основам кинематики элементарных процессов Учебное пособие
Подп. в печать 15.12.2010. Формат 60×84 1/16. Бумага офсетная. Печать цифровая. Тираж 40 экз. Заказ Т-279. Отпечатано с диапозитивов, предоставленных автором, в типографии «КДУ». Тел./факс (495) 939-44-91; www.kdu.ru; e-mail: [email protected]
© МГУ, 2010. © НИИЯФ МГУ, 2010. © Строковский Е. А., 2010.
ISBN 978-5-91304-154-8 © Издательство КДУ, обложка, 2010.
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9 C 1 7 /1 1 0 / / F10 1 C B GO 1 25
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A1
A2
A3
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). F""AG
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− sinh η 0 0 cosh η
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⎛⎜⎜⎝
A0
A1
A2
A3
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",
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P2tot = (E∗
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proj +m2targ + 2mtarg · Eproj =
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√s F$
0 12 12 1 a + b → c + d + X ? 1 72 E∗
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"
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√s N 1 72
FG 0 1 / 72/ 1 95 0 1 9/N 0 1 0 /
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10
100
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1 10 102
103
104
105
106
107
108
109
pp
10
100
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50
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10–1
1 10 102
103
104
105
106
107
108
109
⇓
pp_
Cro
ss s
ecti
on (
mb)
Laboratory beam momentum (GeV/c)
Cro
ss s
ecti
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mb)
Center of mass energy (GeV)1.9 2 10 100 103 104
total
total
⇓
elastic
elastic
H " >2 1/ 1 1 1 Q *R
A
10 1 10 10 10–1 2 3
10 1 10 10 10–1 2 3
p_nelastic
p_ntotal
⇓
pd
pn
Cros
s sec
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(mb)
Laboratory beam momentum (GeV/c)
Cros
s sec
tion
(mb)
1.9 2 3 4 5 76 8 910 20 30 40
2.9 3 4 5 6 7 8 910 20 30 40 50 60
pd total
pntotal
npelastic
Center of mass energy (GeV)
10
100
1000
100
200
200
2000
500
500
10
20
50
20
50
5
⇓
p_dtotal
H >2 1/ 1 1 Q *R
B
⇓
πdπp
Cros
s sec
tion
(mb)
Laboratory beam momentum (GeV/c)
Cros
s sec
tion
(mb)
Center of mass energy (GeV)
1.2 2 3 4 5 6 7 8 910 20 30 40
2.1 3 4 5 6 7 8 910 20 30 40 50 60
10 1 10 10 10–1 2 3
10 1 10 10 10–1 2 3
10
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Laboratory beam energy (GeV/c)
Cros
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Center of mass energy (GeV)K −
−
d
K N 1.6 2 3 4 5 6 7 8 9 10 20 30 40
2.5 3 4 5 6 7 8 9 10 20 30 40 50 60
⇓
K –pelastic
⇓
K –ptotal
K –nelastic
K –dtotal
K –ntotal
H B >2 1/ / 1 0 Q *R
E
0
2.5
5
7.5
10
12.5
15
17.5
20
22.5
25
–1 210 1 10 10 10
3
K +ptotal
K +pelastic
10
15
20
25
30
35
40
45
–1 210 1 10 10 10
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K +dtotal
K +ntotal
K +d
K + N
Cros
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(mb)
Laboratory beam momentum (GeV/c)
Cros
s sec
tion
(mb)
1.5 2 3 4 5 6 7 8 9 10 20 30 40
2.5 3 4 5 6 7 8 910 20 30 40 50 60Center of mass energy (GeV)
⇓
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H - >2 1/ 1C / 1 0 Q *R
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10-4
10-3
10-2
10-1
1
1 10 102
⇑
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Center of mass energy (GeV)
Cros
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Laboratory beam momentum (GeV/c)
γp
γd
0.3 1 10 100 1000 10000
0.1 1 10 100 1000 10000
10
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10-1
1 10 102
Laboratory beam momentum (GeV/c)Cros
s sec
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(mb)
Λptotal Λpelastic
10
10 2
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1 10 102
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Center of mass energy (GeV)2.1 3 4 5 6 7 8 10 13
Center of mass energy (GeV)2.1 3 4 5 6 7 8 10 13
γptotal
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H E >2 1/ 21 1 . C / / Q *R
,
*,&9 4(. 3 :%34,42 ,2 ;942+,29
H 0 1 / "N - F 0 9 / 1 G . 1 +
0 1 2 3 4 5
10
100
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[mb]
η
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2 0 10 0 1C % 1 ” 2” C 1
*
1 2 15 b12 = −(u1 − u2)20 2 uiNB " 0 950 C B F#$%&'()*+'G 9 F),$-')G0 btp0 0 1 9 γp = 1+ btp/2 .5 btp C C F F""EGG
; 0 2 7 / 0 C / / 720 / 1 0 71 5 ;?
• 72 FC 12 C 10 29/ G0 2 9 9 "O 7 /9 1 2O
• 1C F1/0 G 72? 912 F10 1C 2G0 C12 C 0 C6/ C 2 2 1O 7 F/G 2 0 / 7 / 720 2 9 C " FABG
• 72? 0 9 / 2 1/ @ 9 FABG C 9 FC 0 6 7 1 ””0 C 0 10 12 C J22G
<:% := 7+86
7 /9 1 1 1 0 7 5 / 110 6 C 2/
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F G 9 0 / / 0 2 C
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<:% 394,= 7+86
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0 71 7 5 9/ 0 / 17 C C5 ”1/” 11 / / 1 >1 C 20 5 F G C 0 5 C5 9 ; 71 0 11 5 1 0 N 56
A"
C95 J 7 1
0 71 5 0 10 5 5 5 0 71 / 72 / 9 1 / / / 0 F1 0 / 19 G
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• < 1 0 1 1 / / / K F 7 / G I 1/ / 1 7 1 / 72 1C / 72 C 2 . 7 1 1 2 ” ”
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I 0 1 1C /F1//G 72/ 1 $J 6? / / / / > $J 5 "O C 71 0 55 ; 1 / 6 5 1 1/ 72
1 1 72 0
A
7 C 6 0 / /0C 1C/0 12 F1G C72 / /
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!"
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. 1C / 72 1 C / 72 1 9 1 0 1 C 7 / 2 2 5 / %95 2 1/ 3 1
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= / C / = 1 C 1F C / G
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• ”> ” 0 N 1 C O 21O C F10 ∆CG FηGO 0 12 C ? 2 K 1
• . / 1 0 25/ 8 2 6 C0 1 1/ / C
• < 1 ”/” ”/” / C 2 / 1
• H 1 / 2 1 >1 1/ 5 1 /0 1 5 F∼ 0.8 =G0 / C
•
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• / C / 12
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8 7 1 8 20 72 C 1/5 / / 720 C 0 C 1 9 10 6 /C5 B
C 7 10 1 0 / / 72 1 0 2 / 720 7 5 2 C 0 9 /1 > C 5 2 56
1/0 2 7 ” C”0 1 2 C / 1 / / F 0 21G0 5 0”5 ” 0 C 2 τ0 0 1 1 1 γβcτ00 1 1 2 ; 0 C 1
A-
1 6 0 7 C 7 C 1 0 2 2 2/ 0 C 2 1C
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• F29/ 15 G 1 / 1 F 71 ” C” GO
• 21 F ” C → 11 ”GO
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<:% +&.23424= 2.-&9= ,,3
; 56 F$JG 2 1 C/ / 720 C 6 1/ 12
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2 1 0 C 0 6 5 56 562 $J 6 ? 6 0 1 6 0 25 1 C0 2
. / 72/ 1 F G 1 /
2 9 9/ 1 U/ 1 ; 7 C 161 165 C/ / 72 8 256 2 1 0 6 2 56 F$JG F ,G0 0 2
A+
>5 FG C 1 ” 2” 2 6 0262 21 @ 0 56 2 / 9 / O 1 5 2921
30 1 71 2 56 $J N 1/ 122 C / / 1 F "!G?
" 56 F$JG 0
75 F10 22 1 2 C /0 2 C 2 1 G0
A $J 5U 5 5
H , !" #$% & ! ' ())* !+ ! ' ()), + ! ' ()), -./ 0 . / ! 123&4
A,
H * !" #$% / ! 0 1 ())5 4 -./ 0 . / ! 23&
. C 1/ I 10 2 0 5 9 0 5 0 72 F 0 G
√sNN = 4÷ 9 37 0 10
[\[\ ]]0 C / 86 / 7 / 5 0 1
C / C 5 1/ 1 1 2 / / 1/ 1/ 1 7 / /
10 5 12 0 C 1/ C / ” ” / 1/ 0 12C 0 2 / / 1/ /0 12C/ 565 $J F20 0 1 9 2 G .7
A*
B
Pre-cooking by e.m. interaction??Z1+Z2∼160>137; B ∼1015÷1017 Gs
B
B
H "! & 3 !"#$% 6 0 0 1 4 " " ! ! + 0 0 0 . . " 1 0 1 4 ! 4
56 6 ”2” ? / 20 1 C 2 0 1 U 9 O F / 2 G 75 $J 6 C 1/ C0 1
C 0 16 / C/ 0 2 7 2 1 25 1 9/ 7 2 / 1 I 15 0 10 1 / / C/ 2 1 9 2 1 F ""G0 1 1/6 2 1 / K
.7 1 C0 1 9
B!
1 122 5 ”1 12 ” 5 F "!G0 1 56 C2 0 56/ 0 5 F 2 G 15 F 1 // 2 /1 2 2 / 1 G$ 7 2 1 7 56 $J 0 1 75 1 // 7 2 / 10 1 % 20 0 1 1 1/ 5 7 2 / F 7 2 Q "RG
H "" $ .7
B"
t i m e
1970’
1986’
1996’1996’
2008’
2015’
H " 8! 1 4/ 98:8 ;<=1>?@4 @8#- 198:84 ABCD 1EFC4
I 2 C/ / 72 6 1 5 56?
• 1 FG 0
• 1 0
• 1 / 1/ C $J F G0
B
• FG 1 / 1 /6 720
• 1 C/ 2 U 0
• 7 / //7 2 / 1/0 56/ 1 / C/
C 5 C 0 71 / / 2 1 0 56 1 2 9 F /G C9 2 2 1 C/ / 72 7 /
> 1 0 / 8;<; 6 / F " G 1C56/ 2/ /
BA
8 7 / N / U 1 $ 10 1 72/ 1/ C 2/0 2 F 0 G 2 F1 G 2 6 C 0 71 1 1 C 1 F 0 0 1 1 5G0 2 / 0 / / 0 15 /
H 5 / a b0 0 5 c d F 2 2 G 211 0 / 2 20 7 211 5 1/ 1 72 / 1 1 1?
a+ b→ c+ d+X , b(a, c)dX b(a, c) . FA"G
B-
. 1 7 / / 72 1 / 1C / 72 8 O / 5 ? 1 0 11 N 2 71 O 1 90 1 N 2 0 /0 0 X 5 2 5 2 2 0 1 B1 PX 7 PX 0 MX =
√P2X "
"? PX / / 7210 12 B1 /6/ 7
%& &&
>! > 1 9 " F 0 5 9 G . 12 5 1 1 0 2 / F0 1G 5 8 12 1 0 F/ G ! I 1 5 ” 7 ” F ” 7 ”0 10 1 1 /G0 2 122 C F 9G 9O 1 7 9 1 ? C 1 / 20 2 12 1 H / C C 5 0 / 2 9 I 0 12/ 12/ 0 16 F G 90 C6 F C G . 7 / 1 0 2
BE
2 F N m2 = E2 − p2G02 / C .7 2 C 0 / ”7 2” C / 12 0 2 ? 1 0 2 7 F C ”7 ”9G C56 2 C1
?! ) + > 1 1 1/ 9 2 1O C C C ; 5 7 2 10 10 1 9 90 2 C5 0 ” 2 ”0 2 5 C 2/ 2 0 8 2 2/ F10 G 8 1 5 / / / 29/ 7 / 7 C 1 C 0 1 9/ 0 / 905 ”2 ” 9
@ 6 6 > 1 12 1 / 72/02 C/ ? 2 90 A 7 1/0 / 56/ 0 9 20 C 1 5 C 2 2 2 0 / 10 1 C 1 1/ 1 /
4&% 3&% % 3 I 1 1 1 C 2 C$ 3&%' 71 C 1 0 / 2 5 1 0 / 0 / / 0 &0 1 71 0 / 2 F G0 1 O / / 5
B+
10 7 6 C 1C / / 72 71 1/ 5 1
42 0 71 / 52 1 2 7 / 0 2 75/ 71 / 7 I 0 C2 C0 1 ? ”5” ”75”
'( "&
*+9 %/&542%%
$ C 1 s0 56 1 72 0 C9/ 1/ 1 t0 12 B1 / 10 uO 7 / 1/0 s0 t0 u 0 /
L 1 / 0 5 5
a+ b→ c+ d , b(a, c)d FA G
12 0 2 c d C 2 a b0 5 0 6 20 2 1 720 1 / 721 . Pi N B1 122 1 FA G ;
s = (Pa + Pb)2 = (Pc + Pd)
2 C O
2 0 6 0 t = (Pa − Pc)
2 8 ! a (b + d) 80 1/ C 1 12 b (a+ c) B10 1
t = (Pa − Pc)2 = (Pb − Pd)
2
B,
I 0 C 1 Pa0 Pb0 Pc0 Pd
u = (Pa − Pd)2= (Pb − Pc)
2O 2/ /
% 20 FA G C 7 5 ?
s+ t+ u = m2a +m2
b +m2c +m2
d . FAAG
29 1 1/ 0 0 5 / 721?
Pa + Pb = Pc + Pd .
7 2 1 s. Pb 15 0 Pc 5 12 0 1 t 0 Pb 1 15 Pa0 Pc 5 1 Pd 12 0 1 u 2 1 10 1 562 / B10 1 2 B 250 B 1 1 F ” ” G0 56
0 " a + b →1 + 2 + 3 + ... + n ! 12345 (n − 1) !
(2+3+...+n) ≡ X " "M2eff ≡M2
X = (∑n
i=2 Pi)23
> 0 2 X / B1
L 1 1 t0 12 50 2 FA Gma = mc0 mb = md . t 0 FA G F7 0 5 1 F 0 / 72G1 12 0 5 11 / 1 95 15 1 /G 16 0 2 7 0 0 2 ϑ F 2
B*
6 CG0 1C sinϑ ≈ ϑ /9 0 0 2 1 ∗ ;?
t = (Pa − Pc)2 = (E∗
a − E∗c )
2 − (p∗a − p∗
c)2 =
= −2 (p∗)2 + 2 (| p∗ |)2 cosϑ∗ = −2 (| p∗ |)2 (1− cosϑ∗) =
= −4 (| p∗ | sinϑ∗/2)2 − (| p∗a | ·ϑ∗)2 − (p⊥)
2. FABG
F@ 1 0 11 1 p⊥ p∗a · ϑ∗
1 1/ 0 17 1 FABG C ∗G
; FABG 0 FA G 1 | t | 0 12112 1 2 ϑ 1 / 9 1 320 2 0 t ! 6 " 7 6 ! 8 6 | t | !
.0 1 1 9 99 1 0 1O 10 / C 0 1 1 0 N
; 1 0 2 1 0 1 9 5 5 R0 C
f ∼ R
2 sinϑ/2· J1 (2kR sinϑ/2) =
Rk
2k sinϑ/2· J1 (2Rk sinϑ/2) . FA-G
@ J1 % ; FABG0 2k sinϑ/2 √−t 1
f ∼ Rk√−t · J1(R√−t) . FAEG
10 1 1 0 0
dσ
dΩ∗ ∼| f |2∼R2k2
−t · J21
(R√−t) , FA+G
-!
1 dΩ∗ = dφd cosϑ∗ = dφ2k2d cosϑ∗/2k2 = dφdt/2k2 FFABGG0 FA+G 0
dσ
dΩ∗ =k2
π
dσ
dt∼| f |2∼ R2k2
−t · J21
(R√−t) . FA,G
. 2 1 t0 1 dσ/dt 2 5
dσ
dt∼ πR2
−t · J21
(R√−t) . FA*G
; 0 * * ! )! ! + ! 0* *% '7 * 3! F 20 7 GO 2 1 11 0 1/ > 0 2 F FA-GG
. 7 5 0 5 11 2 0 0 7 / 6 0 0 C C 0 1 / 122 7 / 1 t C / 0 FA*G ; 7
*,+,8 +%6
@ / 72 1 0 2 9 1 9 / ; C0 C 1
Pinit = Pfinal , (Einit,pinit) = (Efinal,pfinal)) , FA"!G
2 Pinit N 1 B1 20 Pfinal N 2 0 E p N 1 72 1 56/
! ""
-"
; C0 1 72 0
s ≡ (Pinit)2= (Pfinal)
2. FA""G
F7 C 1G / F10 nG 1
F C C 1 G 1 " 0 1
M2eff = P2 =
(n∑
i=1
Pi
)2
= FA" G
=
(n∑
i=1
Ei
)2
−(
n∑i=1
pi
)2
≥(
n∑i=1
mi
)2
,
2 mi N 7 / n 2F 2G 0 s M2
eff 15 9 FA" G 0
b(a, c)dX F FA"GG
s ≥ smin ≡ max(ma +mb)
2, (mc +md +MX)
2, FA"AG
2 smin N s0 1 FA"G 6 9 / 721 8 1 ! FA"G <0 C 12 72 F aG C 9 2 0 2 1 FA"AG > 72 !" !"O 1 7 5 5 725Tthresh = Ethresh − ma 10 T 1 720 E N 172 E = T +m F 2 1G
!;
H 2→ 3 F A"G0 10 125 5 1 1 2
C 0 5 pp→ pp+meson0 125 5 725 Tthresh 1 C t 120 1 B1 F^_`ab:cdebG 1 2
-
H A" #0 pp → pp + meson 0 !0 2 → 3
/C Tthresh 1 1 12 s 0 1 0 s / 1 1 56 C?
Tthresh = 2m(1 +
m
4M
),
tp→p = (Pa − P1)2= −mM , FA"BG
tp→meson = (Pa − P2)2 =M2 ·
(1− 2 · m
M
).
C 6 7 1 pA →p+A′+meson0 2 A′ F 0 11 C G 3 11C0 9 b C0 3 C 0 1 ; 0 0 0 a 10 56 1 1C2 A"0 5 2 0
T threshprojectile = MX +
MX
Mtarg·(mproj. +
MX
2
),
MX = M3 +m2 −Mtarg , M1 =Ma . FA"-G
8 5 0 9 90 9 12 F1 1/ / /G MX C 0 1 C20 ”C” F ”1 ”G F C A G
-A
C pp F 211 G F5 1 pp→ p+Λ+K+G0 3 FGMY 0 FG mi0 i = 1, ... I2 C 0
tp→p = − Mp
Mp +MY +∑mi·[(MY +
∑mi
)2−M2
p
], FA"EG
tp→mesons =M2p ·[1−
∑mi
Mp
(1 +
MY
Mp
)], FA"+G
tp→Y =M2p ·[1− MY
Mp
(1 +
∑mi
Mp
)], FA",G
2 tp→a0 Fa = p , Y, mesonG 12 C a B1 F C AAG
; FA"BGFA",G 0 ” C” C 0 9 F1 5G 1 B1 8 0 1 1 9 ∼ 1 372U:2 / / / 1 7 / I 12 / 72 F@ 0 0 9/ 1 %922 $ FfXYG0 2 12 J22 F 6 G ; 0 // F1 192 G 72 C 120 112 G
8 ” 12” pp ?
(2M +MX)2 ≤ spp < (2M +MX +mπ)
2. FA"*G
”.12” pA F A G C 2 0 9/ 2 F A G 2 F10 1 2 1 2212/ A G?
4M2
(1 +
MX
2M
)·(1 +
MX
2Mtarg
)≤ spp < (2M +MX)
2. FA !G
V ”112” ?
sthreshpp − s threshpp = 2M ·MX ·
(1 +
MX
2M
)·(1− M
Mtarg
). FA "G
-B
0 2 4 6 8 10 12 14 16 180
500
1000
1500
2000
2500
3000
3500
4000
положения деполяризующих резонансов(Нуклотрон, дейтроны)
Ξ+2K
Пороги, достижимые на ускорителях протонов(промежуточные энергии)
φ
Ω+3K
DC12C d p
COSY
Nuclotron
рожд
енны
й в
реак
ции
избы
ток
масс
ы, М
эВ/c
2
кинетическая энергия пучка протонов (в л.с.), Tkin, ГэВ
H A %! 0 pp p d p 12C -./ . 0 3 !0 1 !0 1 4 3 ! 03 0 !4 G ! 0 3 ! =HFI 1 4 @! 1 4 2 . ! . !" @! d !0 1 .! JK))L4 0 . 1 ! 04+ 0 + . ! " 3 !
.12 0 / C 161 7 C /0 7 0 2 F G 0 1 C ”9” 9 2 3 20 / 2 F 1 1 2 20 2 / 10 F G
--
H AA M0 *! 0 pp
2,42&5%. 4,+,425
16 1 1 1 m? u = P/m0 2 P NB1 7 1
H S S ′0 1 S ′
C S B 5 u = (γ, γβ) 1
Z0 β = (0, 0, β) $ 20 β = v/c0 γ = 1/√1− β20 c = 1
H AB V / C 9 / 5 β F1 1G
L & C 10 1 1
-E
9 / S F ABG 9 /50 1 B A = (A0,A) ≡ (A0, A1, A2, A3) ≡ (A0, Ax, Ay, Az)15 2 F"""G?
A′0 = γ(A0 − βA3
), A′1 = A1 , β =
v
c
A′2 = A2 , A′3 = γ(A3 − βA0
), γ =
1√1− β2
9 ξ F 21 G 1 F""EG?
β = tanh ξ , γ = cosh ξ , β · γ = sinh ξ ;
1 F F""+G0 1 / 1 /1// S S ′ S ′′0 2 S ′′ C S ′ C 1 Z0 1 S ′ β1 F SG0 S ′′ β2 F S ′ β3 SG0 / ?
ξ3 = ξ1 + ξ2 ,
1 1 1/ 1/ 2 21/ 4 5 0 βi 0 1 0 7 20 0 0 2
8 0 165 1 B C 1 6 1 4 1 2 1 0 1 1 QA!R F 0 2 10 / /G
0 1 2 1 0 10 10 5 1 2 0 C1 B 01 0 B u1 u2 B 0 1 15 B 0 uµ u
µ 0 ? uµ u
µ = 1 . 1 2 S ′0 / S 9 B u1 u2 42
-+
0 B 1 1 20 B u12 0 ?
u012 = (u1 · u2)u12 = u1 − u2 · u
01 + u0121 + u02
. FA G
F 0 u01 = E1/m1 = γ10 u1 = p1/m1 = γ1β1 G
C 0 6 B [0 12 S S ′0 2 9 / C ' B 5 u0 1 F"""G C 1 ?
A′ 0 = (u ·A) FA AG
A′ = A− u · A0 +A′ 0
1 + u0. FA BG
FI 1 2/ / O 10 1 12/ MH 1 4G
3%+%29 7+8 ?10&549@ (%42
. s 0 C 20 1 72 56/ 0
s = (Pa + Pb)2= (E∗
a + E∗b )
2, FA -G
2 / 0 20 1 $ 20 p∗
a = −p∗b 10 m
2i = E2
i − p2i 0 C0
E∗b =√s− E∗
a , FA EG
1 / ?
E∗a2 −m2
a +m2b = s+ E∗
a2 − 2E∗
a
√s ⇒ E∗
a =s+m2
a −m2b
2√s
. FA +G
;1 p∗ C a b ?
p∗2 = E∗a2 −m2
a = E∗b2 −m2
b =
(s+m2
a −m2b
2√s
)2
−m2a , FA ,G
-,
1 2 1 / 10 1 C5
p∗ =λ1/2
(s,m2
a,m2b
)2√s
, FA *G
2 λ(s,m2
a,m2b
)N 0
!" " . 1/C F !G 7 2 2 0 1 95 3 16 2?
λ (a, b, c) = a2 + b2 + c2 − 2ab− 2bc− 2ac . FAA!G
1 7 0 / 2 1 0 1 FAA!G N L 56 1?
λ (a, b, c) = (a− b − c)2 − 4bc . FAA"G
0 1 p∗ F C 9 G0
√s ≥ ma + mb
F FA"AGTG0 1 7 2
L 9 FA *G FA +G0 1 1 1/ 72// 2 c dD 42 2 0 N 20 ma0 mb mc md 0 C0 C ;? FA *G 0 112 a + b → a′ + b′ 1 0 C 1 2 ϑ∗ . 0 9 C C 1 Fp∗⊥, p
∗‖G0 2 p
∗⊥ N 11
F 11 15 2 1 G 1 0 p∗‖ N 1 1
F1 1 15 21 G 7 1 p∗ C C p∗
C 12 Fmc = ma mb = md0 1 G 72 9 562 120 1 2 C 22 p∗fin ≤ p∗ F 12 7 C C G
-*
A&&14,/ 10&54,3 8, 1+
; 0 FA *G 1 1 Fp∗⊥, p∗‖G 2
9 1 2 .0 0 5 C F /0 10 c0 1 2115 1 B1 Pd 7 F TG m2
d = P2G0 A1 c C 2 2 7 9 $ 2 2 9 / 0 / 1 D F 7 20 0 G
<0 1 1 4 Fγ βG 1/ 5 F 5G C C 0
βcm =pa
Ea +mb≈ 1− 2m2
b
s, γcm =
s−m2a +m2
b
2mb√s
≈√s
2mb. FAA G
; 0 1 1 c 2 @ C0 1 c C C p∗fin0 p∗fin C 2 FA *G0 1 1 . 1 1 1 z F 0 0 C 1 7 15G @ 0 1 c 2 9 1 4 " 1 1 p∗
c 0 1 7 2 1 / 1 1 5 @ 0 2 9 0 1 14 F15G 0 0 1 F1/ 5 G 1/ 71
2 10 1 pc c C C6 30 0 1 C0 7 71
E!
0 s C 12 0 C C 2 9 F C 710 C2 G0 9 1 !
<0 1 71 1 p∗c 8 2 95 1 1C @ 0 112 1 p⊥ 6 0 6 1 5 2 1 1 .7 9 C 1 (p⊥, pz) 71 3 1 1 7
1 1 40 F p∗ ≡ p∗cG?p⊥ = p∗⊥pz = γcmp
∗z + γcmβcmE
∗ FAAAG
E = γcmE∗ + γcmβcmp
∗z ,
1 (p∗⊥p∗
)2
+
(p∗zp∗
)2
= 1 . FAABG
9 FAAAG C 11 ?
p∗z =1
γcmpz − βcmE∗ , FAA-G
1 20 1 7 FAABG 1 9 FAAAG0 2 1 71 1 (p⊥, pz)? (
p∗⊥A
)2
+
(pz −HB
)2
= 1 , FAAEG
2?A = p∗ , B = γcmp
∗ , H = γcmβcmE∗ . FAA+G
$ FAA-GFAA+G0 C FAABG 2 ? 1 1 2 γcm 0 11/ 0 2 γcmβcmE
∗@ 1 ? 2
15 0 / / 1 72 . C 7 / 1 C 2/ Q"0 0 A0 BR
E"
8 C 0 / 0 ? C/ 71 FAAEG 8 C 1 1 71 C / 1 71 0 C 2/ 1/
8 9 0 5 0 β∗ = p∗/E∗ H C/ 11 ? 2 p∗ = (p∗
⊥ =0, p∗z = −p∗) 2 p∗ = (p∗
⊥ = 0, p∗z = p∗) . 1 5 1?
p1 = (0, γcmE∗ (βcm − β∗)) , p2 = (0, γcmE
∗ (βcm + β∗)) . FAA,G
H A- $ 71 1
80 p1 p2 pmin pmax 1 0 1 A C/ F A-G?
" βcm < β∗? 1 1 pmin
1 1C 15 1 F ””0 1 2 ",!GO
E
βcm = β∗? pmin = 00 1 O
A βcm > β∗? 1 1 pmin
1 1 1 F ”1”0 1 2 !G
$ A-0 / 6 2 2 C 1 1 Z ? G 7 2 *!0 AG 9 *!0 1 52 2θ < θmax C/ 1
0 10 725 1 1 725 0 19 90 1C5 FAAAG?
E∗ = −γcmβcmp cos θ + γcmE , pz = p cos θ . FAA*G
8 1C 2 10 p0 2 0 5 12 1 A- 0 FAA*G ?
E∗ + βcmγcmp cos θ = γcm(p2 +m2
)1/2. FAB!G
> 9 C p(θ)
H9 C 1 56 Fp± 9 FAB!GG?
p± = m · βcmγ∗ cos θ ± (β∗2γ∗2 − β2
cmγ2cm sin2 θ
)1/2γcm (1− β2
cm cos2 θ)FAB"G
0 C 0
p± = p∗ ·cos θ
(g∗ ±√D
)γcm (1− β2
cm cos2 θ), FAB G
D = 1+ γ2cm
(1− g∗2
)tan2 θ =
β∗2γ∗2 − β2cmγ
2cm sin2 θ
β∗2γ∗2 cos2 θ, FABAG
2
g∗ =βcmβ∗ . FABBG
EA
1 72 E± =√(p±)2 +m2
8 0 D = 0 5 g∗ = 10 β∗ = βcmO 2p+ = p− . g∗ < 1 p− F 1TTG 0 0 9p+ . g∗ > 1 90 p− p+ 5 C 1 0 7 C 2 1 9
sin θmax =γ∗β∗
γcmβcm=
γ∗
g ∗ γcm . FAB-G
0 C 2 8
tan θ =sin θ∗
γcm (cos θ∗ + g∗). FABEG
*+/42%3& , 00&.239= 1+,44%=
H C 2 1 12
AE 1 1 71 1 1 2 C NN dN / F 56/ 71 1 G . 1 1 p =
√p2L + p2T 0 2 pL pT N 1 F
1 1 G 11 1 > 71 25 F / 721G C 1 10 C2 / /I 1 0 9 p + p → p + p + π0 0 1 7 0 9 d+p→ d+p+π00 2 ” ” 0 F G / HC 1 10 1C6 7 0 6&
; 0 7 8 07 6 07 9 ' 3!0* %' ”0”
10 1 01 C 15 C 0 16 /NN
EB
H AE M 3 ! p(p, π)X p(d, π)X 9. ! 3 ' p(p, π)X 1. ! .4 N . 3 ! p(d, π)X ” 0” 3 X 1 !0 .49. ! . 3 ' . " 3 ! 2! ! 3 ' ”!! ” . ! p(p, π)X 3 !9 0 ! p(d, π)X
F@ 1 ”” 15 7 / 1 G
/ 1 2 10 1122 C / 5 F10 512 1 / 72/ 10 2 11 C 0 1 p d A G
+08 ”1,+,8” 01+08= +%6
120 C 1 2 H 5 1 2→ 20 20 10 1 ”1” 9 16 250 9 7 1 1µ+m→ µ+M 0 2 µ N 0 m N 90M N 0 5 1 1 9
E-
; 16/ 0 2 2 0 C 72 2 9 2 12 0 C 2 ”12”0 ? 1 B1 C F1 5G C 12 | tmin | F10 122 C 5G 565 F !G
EE
!"""
!& "( )
8 1 / 72 1 C 9 H =0 1 "*E* 2 Q+R
2 7 0 C 2 1 C 56/ 2 2 ”” 56/ C 0 ”1 ”0 56/ C 0 56/ 111 F1 1 1 112 1 p⊥ pi 15 !G0 1 7 C 1 1 5 F (x) 5x 12 1 P0 ; F10 G0 56 x0 1 95 F 0
E,
1 9G I 0 / 1 7 / / / 1 2 20 1 1 1 x0 / F (x) ”8” 1 2 0 95 190 1 0 1 / 0 2 7 / 11C 1 / C 0 2 1 9 ””1 2 ? N 1 0 F”5 ”G0 1/ 2 1 0 56 16 1 0 /
C 1 25 C 1 2 0 2 15 /0 1 F B"G H1 1 1 F 1 xG 1 Ψ(x)
80 1 5 0 1 ? / 9 1 F 2 1 G 2 C > C0 10 0 11 1 2 F 0C 11 1 C 0 p⊥/M 10 2M N G0 0 21 2 1 &
!& )
8 1/ 95 1 0 1C QA-R > 1 2 QAER / / QAER QBAR 6 1 7 / 1 1256 1 5 1
E*
H B" =2 FG / FG 1 F1 G .1 1 1 0 1 F2 2 C GC C 1
H B =2 / 1
/ 0 10 2 1 C 1 7 /1 1/ 2 0 =
. 5 5 7 2 1 0 C6 1 Z 1 (t, x = 0, y =0, z) 0 t = 0 1 (0, 0, 0, 0) 7 C 1 (t, z) 1 F B G0 C6 2 0 C2 B 10
+!
6 1 2 ±45 Z $ 0 C 0 C 56 2 2 0 ? 2 1 0 1 5 t0 t
ψ(t1, z1) F G t = t10 ψ(t2, z2) t = t2 C 1 0 1 1 75 ψ(t1, z1)O / ∆t = t2 − t17 1 75 2 56 2 0 2 1 ψ(t, z1)0 B 1 211/ t = t1 ψ(t+∆t, z2) 211/ t2 = t1+∆t $2 C Z 0 ”1C ”
I1 1 0 0 ”1C ” 0 755 C 1 2 0 1 1 56 56 2 8 7 (τ, ξ) (t, z) F B G
τ =1√2(t+ z) ; ξ =
1√2(t− z) FB"G
. QA-R 10 1 1/ (t, r) (τ,ρ) 1 5 7 1 0 2 1 1 56/ 1/F@ 1 (x, y, z) r = (z, r⊥)0 56 N ρ = (ξ, r⊥)G 0 2 C6
H =p2
2mFB G
/ / 1 F10 QAERG
H =p2⊥ +m2
2η; η =
1√2(E + pz) . FBAG
# $ " " % # $ " " %&" $
+"
2 C FB G FBAG F 1 η 2 1 G 2 0 1 C 1 11 / 0 11 F1 95 15C G 1 F G; 7 K 1 1 1 0 F1 / 9 /G C 1C0 /0 1 1 22 0 1 22 / / 2 0 6 20 5 1 5 5 O 1 p⊥/p‖ 1 1
% 1 2 1 1 / / / 7 / O 12 56 /? QA-RNQB!R0 C QBA0 BBR Q--RG
*
(0 0 2 N N "*A 2 S0 % $ 1 29/ 0 5 0 2 QA R
- 1 / 261 1 1 2 2 1 2 2 1 FC6 1 1 0 2/ 2/ / G . 1 N C "*AA 2 QAAR F> V O "*AB 2 H0 $2 @/GO 2 2 1 1 "*AE 2 F$20 H @/0 5 G 2 2
'( & $ ( ( ) ( $( µN = 3.15254166(28)×10−14 *+,-#
+
Fµd = 0.85742 · µN G 2 / Fµn = −1.9130428 · µN G 1 Fµp = 2.79284739 · µN G?µd ≈ µp + µn . 1 0 7 0 / 0 0 1 F 5G 8 50 11 .0 0 1 5
%19 1 C 1 d+d→ α+π0
F d(d, απ0)G 0
R =σ (p+ d→ π+ + t)
σ (p+ d→ π0 +3 He)≈ 2 , FBBG
F 11C 1 "U 2AG C 2
$1 1 "*A* 2 2 1 F$20 H0 H @/ QABRGO 2 2 0 0 1 D 2 F1 G
+ " "( )
"
1 1 / 1 "! 37U 1 1 / F " 37G / 72/ 0 10 / 122 0 2 1 1 2 1/ % / 0 71 C . 7 0 10 1/ / 7
1 1/0 0 0 2 1 1 0 71
+A
H BA . 9 F15G 2 1 F2 G 1 / 122 d (p , d) p ”” $ / / / ? 12 0 2/ 0 C 5 C 2/
1 % / 720 20 1 1/ / / F QBE0 B+0 B,RG
7 2 1/ F 1 16 G C 1 165 15/ 20 C/ BA
20 1 5 BA0 / 1 = ;? 10 2 9 / 1 FG . 2 1 0 C 7 / 1 1 5 5 12 1 1 1 1 1 1 0 /0 1 1 F/ 9 2G H 1/ 20 2 F G 16 95 2 7 2 ” ”0 1 12 5 N ”1 ” 0 2 10 0
+B
0 7 1 0 / 2 0 2 11 1 1 F 1C 1 G0 1 ”1”0 1 1 1 1 1 0 7 1 9 1 1 F 0 56 95 1 0 1 0 1 1 G I 2 2 ”” 0 1/ / 1 0 0 7 1 1 2
α =p‖ + Espect
pd + Ed, FB-G
2 p‖0 Espect N 1 1 1 72 1 0 pd0 Ed N 1 72 F G
= 1 1 1 / 6 2 F1 2G 11C 8 1 C 71 Fp‖0 p⊥G 2 1 1 N 1 k
/ 7 11C 11C ” ” 2 0 )+ 2 k 1 1 C
k⊥ = p⊥ ; kz =
(α− 1
2
)·√m2
p + p2⊥α (1− α) ; k2 =
m2p + p2⊥
4α (1− α) −m2p , FBEG
F2 1 0 1 1 G0 k C0 /O ) + 1 1 2
+-
1 p⊥ ∼ 0 C0 /
/ ” ” 1 k ” 2 1 ”O α 0 ∞0 1 / 12 1 B1 t B 1 C C 15 2 α?
t
m2p
= 1− 4 (1− α)(k(α)
mp
)2
= 1− 2bdp ; bdp = 2 (1− α)(k(α)
mp
)2
.
0 2 / 1/0 12 72 F 7G0 / 1 I 0 15 mp mn0 0 / / /2
1 C 2 11C0 ? C 1 k 1 1 1 0 2 1 q 8 5 0 156 C 0 1? C2 1 1 1 0 2 1 0 2 F G /721
q ≤ qmax =3
4mN
FmN N G0 2 1 0 2 /0 2 C 0 ∞ 2 0 56 1 1 7 2 11C .7 9 C 7 12 1 21 ” ”
. 21 ” ” 10 C62
. " $ $$ , + */ $ $ p⊥ = 0
+E
1 pd Md0 1 C 1 k 1 11
|ψrel (k)|2 d3k = |ψnrl (k)|2 · 1
4 (1− α) ·√m2
p + p2⊥α (1− α) ·
d3p
Ep, FB+G
2 ψrel (k) N 0 ψnrl (k)N 2 @ C 1 FB+G 1 |ψnrl (k)|2 0 1/ 1/ (kx, ky kz) 1 (px, py pz)02 p 1 1 1
1 2 2 1 F 9 11 1G 1 9 1 2 F1 G 1 5 1 20 1 BA0 2 C F1CG 1
Epd3σ
dp≈ σ (sn, t)(n,targ) ×
× |ψnrl (k)|2 · 1
4 (1− α) ·√m2
p + p2⊥α (1− α) · R (n, d) , FB,G
2 σ(n, targ) 2 1 95 F 56 C 9 2 BAGG0
R (n, d) =λ1/2
(sn,M
2targ,m
2n
)λ1/2
(sd,M2
targ,m2d
) FB*G
9 g/ 1 1 95 2 1 ” ” 8 1 6 2 C 0 C σ(n, targ) / 0 2 7 2 1 950 g 1 1 > 1 2 0 0 1 1 1 / 71 2 σ(n, targ) H 2 L4 h&0 6 9 1 R (n, d)
++
= FB,G 1CO 2 BA C0 σ(N, targ) 1 1 95 σtot
(N, targ) 8 2/
1 F 1 / 2G 1 0 σtot
(N, targ) σinel(N, targ)? 1
122 F 1 9G0 7 Meff F BAG 9 9 Meff 9 0 C C 2 10 σinel
(N, targ)
122 F1GO C Meff 1 0 2 BA 16 15 2 7 C 0 565 2 2 122 1 ”” 0 p(d, p )d 1 θ∗ = 180 7 1 C 16 1 2 / 0 σ(N, targ) C 9 20 0 / 9 11 |ψnrl (k)|4
I 0 C A/ / 0 / 1 // 1/ / C 0 1 k $ 20 C C 2 12 1 ””
86 1 2 2 1 2 1 2 0 1 95 1 1 BB 0 C 1 C / 2 FB,G 1 / / F BBG0 / 1 /k ∼ 200 − 500 7U 71 / C 5 C 1 F 1 9 2G C / 2 / | k‖ |≤ 100 7U FB,G 2 1/ 10 C 6 9 0 7 Q-"R
+,
H BB ; 2 1 9 1 2 1 2 0 1 95 1 121 k0 12 >1 1 1 /0 1/ F 1 QB-R0QB+0 B,R 7 / /G 4 N / / 1 9 FB,G F 1 1C0 i&[G 2/ C/ / 7 / / 1 0 1 / / 1 F.C2 QB*R 22 Q-!RG
2 1 / O 5C 2/ /0 1 C7 / 1 1 1 0 1 C 2 F N G
+*
. 1 / 9 2 BA0 2 1 12 5 N ”1 ”
1 2 1 1 F 10 1 1 1 G >2 1C 0 1 /0 0 mp mn0 C 7
1 B1 1 Pp = (Ep ,q)0
Ep =√m2
p + q2 ≡√m2
p + q20 2 q ≡| q |2 mp N 1
I2 B1 0 0 Pn = (Md − Ep , − q)0 7 F Z1 G (Pp + Pn)
2 = M2d 0 2
Md N 2 0 72 1
1 F 16 G?
εp =M2
d +m2p −
[(Md − Ep)
2 − q2]
2Md= Ep ,
εn =M2
d +[(Md − Ep)
2 − q2]−m2
p
2Md=Md − Ep ,
qcm =(ε2p −m2
p
)1/2) = q ,
0 F Z1 G 1 1
1 1 7 2 C F1 Z G0 1 0 1 1 1
B 7 0 0 up = (Ep/mp ,q/mp)O 2 1 1 1 F 16/G?
Ereln = (Pn · up) , qrel
n = pn − up · En + Ereln
1 + u0p. FB"!G
. C/ 0 2 1 ?
Preln =
(Md
mpEp −mp , − q
Md
mp
). FB""G
,!
1 1 0 ” 1/ ”0 Prel
n = m2n0 2 C 0
1 5 > 0 2 F Z1 G C 2 F G1 1 0 2 C
8 2 ? P2n ≥ 0 C 56
2 F 2G 2 ? 1 P2n = 0 2
”12” mn0 1 9 1 1 1 qmax 1 −qmax ; 0 1 720 1 0 1 5 725 1 ”1 ”0 / 72 1 9 .7 P2
n ≥ 0 5 / 0 9 0 2 1 1 C? 1 1 q > qmax0 C ”1 ” 1 5 725 9
# P2n ≥ 0 2
2 1?
q ≤ qmax =3
4mN . FB" G
> 0 11 1 C 9 1C0 1 q 3/4mN 0 2 1 ” 2 ”O 1 1C 7 2 0 1? 6 1 1 F Z1 G0 0 2 /0 1 1 D
9 C 0 2 7 1 C 0 / ” / ”
,"
, % $$$ )
&&
80 2 2 ”212” F3HG0 15 12 1 0 1 0 2 2 1 95 7250 1C 2 1 1 2 FB-G
H B- $ 1 1 2122 1 k0 k ′ N B1 2 2 1 0 P N B1 M 0W N X 0 19 8 N 1C Fγ0 W±0 ZGO 1 B1 q = k − k ′
1 3H 5 Q20 ν0 xBj 0 y . 7 1 /
• . 72 1 1 F17 E0 E ′ N 72 2 2 1 1 G?
ν =q · PM
= E − E ′ ≥ 0 . FB"AG
• $ 1 B1?
Q2 = −q2 = 2 (EE ′ − kk ′)−m2l −m2
l ′ ≥ 0 ; FB"BG
1 C 10
Q2 ≈ 4EE ′ sin2 (ϑ/2) , FB"-G
,
2 ϑ N 2 1 F1 95 151562 1 G 0 5 1 Q2 1 tO 1 95 t Q2 1
•x =
Q2
2Mν; FB"EG
7 1 10 1 1
•y =
q · Pk · P =
ν
E. FB"+G
1 7 720 1 1 1 ; 10 7 1
• $ 7 X ?
W 2 = (P + q)2=M2 + 2Mν −Q2 . FB",G
• 0 C 1
s = (k + P)2 =Q2
xy+M2 +m2
l =M2 +m2l + 2ME , FB"*G
0 1 72 1 / F. 1 G
6 10 5 15 0 2 1 2 h0 /6 5 7 W F B-G?
z =Eh
ν. FB !G
@ Eh N 1 72 2 2 F. 72 ν 1 9G 1 z ? 7 1 720 2 2
,A
<:(46 494& x
. 0 / 1 0 F G 12 1 0 56 5 xp 12 B1 F 1 1 9 ? 2 10 1 9 1G
I2 1 72 2 7 2 1
(xpP + q)2= x2pP 2+q2+2xpP ·q = x2pM
2+q2+2xpP ·q = m2 , FB "G
2 m2 N 1 2 9 1/ / 0 / / 0 5 C 1F G0 2 FB "G
q2 + 2xpP · q = 0 , xp = − q2
2qP =Q2
2νM, FB G
2 0 xp FB "G 0 x FB"EG. 2122 1 7 155 ”% x”0 xBjorken ≡ xB
$ 9 1/3H / C BE
H BE 8 / 1/ @
2Mν = Q2/x
,B
,2,+9= 2+%=
2 ; 81 ”7 ” / / 10 1 ”7 ” 0 1 2 0 0 11 7 1 C 6 0 1 11 F 10 1 7 / / G? 7 n / F0 *!/ 2 G m2
eff = (∑
n Pi)20 2 Pi N B1 i 2
0 n 2 0 0 / B1 > 1 6 O 1 2 / / 1 0 1 0 6 ” ”
A &9 ; $ 7 0 7 1 1 / / 2 1C 1 C 1 8 1/C 1 7 0 1 20 2 2 2 5 F / 1 5 G 81 56 2 ? m2
miss = (Pbeam+Ptarg−∑
n Pi)20
2 Pi C 0 90 Pbeam0 Ptarg B1 9 ; 0 7 0 7 ! %' &%' C F G 1 B1Pmiss = Pbeam + Ptarg −
∑n Pi
,-
# $
%&
”'&(”
, !& xF
. 1 a0 b 1 / 2 c0 1 2 5 F XG0 1 5
a+ b→ c+X . F-"G
L 2C 0 1 B1 F 1 20 72 A1G 2 1 (Ec,pc)0 1 c 6 0 20 56 1 ∗
$ 5 C/ 1 2 C 2 1 0 1
,+
”2 1” F G ”71 1” F 5 2 0 10 G0 01 1 2 C C 1/
p∗‖min = − [E∗ 2max −m2 − p∗ 2
⊥]1/2 ≤ p∗‖ ≤ p∗‖max =
=[E∗ 2
max −m2 − p∗ 2⊥]1/2
, F- G
2
0 ≤ p∗⊥ ≤ p∗ ,(m2 + p∗ 2
⊥)1/2 ≤ E∗ ≤ E∗
max =
=s+m2 −m2
X ,min
2√s
; F-AG
mX ,min N 1 7 X 0 /6/ F 0 1 / / G 2 1 2 10 p∗‖ 1C / 20 1 112 p∗⊥ 2 2 0 2 p∗‖ 15 7 / 0 2 2 1 16 71 10 p‖ 1C / 7101 11 2 1 p∗⊥ 9 0 2 p‖ 15 7 / F -"0 71 1 p(p, π)XG I 7 F G C p‖ F-"G 15 9 7 2 71 F 1 -"0 A0 B C 56/ 1G
1 1 0 C1 0 1
xF =p ∗‖
p ∗‖max
, F-BG
C /9 1 / 5/ 1 F-"G 81 10 . F-BG "" "!" "
,,
H -" ! . p(p, π)X p(d, π)X .! O CC 9. ! 3 p(p, π)X 1. ! .4 N . 3 ! p(d, π)X ” 0” 3 X 1 !0 . 04 9. ! . 3 ' . " 3 ! 2! ! 3 ' ”!! ” . ! p(p, π)X 3 !
<0 1 / 10 1 2/ 1 11 10 1 p ∗
‖max 1
7 12 1 F10 56 B -"G $ 20 1 = 0 11 1 1 0 / 1 1 ”1C ” -"0 1 1 1 1 p ∗
‖max 1 5 2 1 xF 0
?
xF = 2p ∗‖√s, F--G
1 156 F-BG0 2 s 2 9 / 0 /6/ 1 F- G0 112 1 p ∗
⊥ 81 F-BG 1 1 / 72/ 1 /
,*
2 2 1 F C 1 / p∗⊥0 2 12 1 1 2 7 G
, - "
C 0 1 L & 10 % 0 C C 72 1 2 ?
ηc =1
2ln
(Ec + p c
Ec − p c
), F-EG
* %
ηc, long =1
2ln
(Ec + p c ‖Ec − p c ‖
). F-+G
.0 56 0 1 1 C 1 72 ?
p ∗+ = E∗ + p ∗
‖ , p∗− = E∗ − p ∗
‖ , F-,G
0
ηlong =1
2ln
(p ∗+
p ∗−
). F-*G
I 0 6 1 1 F-"G0 5 1 (p⊥ , xF ) (p⊥ , ηlong) 8 1 1/ / 1/ 2 1 > -"O 1 7 C 2 QBR I C 6 0 1 1 / 72/ 10 % 0 1 0 1 160 ? C 10 / 0 1 / 72 0 1 2 θ .
*!
7 1 1
0
ηlong = ln
(E + p‖m⊥
)≈ − ln
(1
2tan θ
),
m2⊥ = p2⊥ +m2
c = p∗+p∗− , F-"!G
F m⊥ ”11 ”G0 1?
" 2 2 9 " 9 ?
m
p θ 1 ; F-""G
11 1 2 9 12?
p⊥ |p‖| ; F-" G
A 1 1 5 ?
p ≥ m⊥ . F-"AG
1 0 1 0
m
p θ 1 , F-"BG
C 0
ηlong ≈ 1
2ln
(1 + cos θ
1− cos θ
)≈ − ln
[tan
(θ
2
)]. F-"-G
. 1C 7 1 1 1 0 156 2 2 ?
ηpseudo = − ln
[tan
(θ
2
)]. F-"EG
0 / / 1 5 1C 2 C 5 1 71 1 / 72/ 15 2 1 1 F 1 QBRG .7 1 0 1 1 1 0 / 1 5 2 2
*"
%0 / 1 C 1 6
0 0 F 1 5 C2G 15 2 C 5 1 F 1 QBRG
H -" 12 1 0 1 1 1 11 0 1 122 1 0 1 / 72/ 11 1 / 1 72 F G; 0 92 71” ” ? 5 1 72 719 0 5 1 C 1? 1 1 11 1 1 1 0 2 11 1 2 9 12 10 1 1 1 1 F1CG 1 1 2 > 0 / 1 71 F G 2 5 / ? C 9 0 C56 0 71 F2 G
, .&( xF
1/ F-BG 1 1 F-+G 2 C 1 1 2 c 8 5 0 7 1 C > ?
xF =sh(η∗c , long)sh(η∗max
c , long). F-"+G
C 0 | xF |∼ 0 1 C? | η∗c , long |∼ 00 7 0 ” ” 0 ∆xF 72 F
√sG C 6
√s F
10 QBRG ; 0 ” 2 ” 5 1 5 | xF |
*
/ 2 C 11 xF 2 0 C / 2 xF ≈ 1 η∗long ≈ η∗max
long ?
η∗long − η∗maxlong =
1
2ln
[E∗max
c − p ∗maxc ‖
E∗maxc + p ∗max
c ‖·E∗
c + p ∗c ‖
E∗c − p ∗
c ‖
]=
=1
2ln
[E∗max
c − p ∗maxc ‖
E∗c − p ∗
c ‖·
E∗c + p ∗
c ‖E∗max
c + p ∗maxc ‖
]. F-",G
. 7 1 15 / 72 0 10 1 12/ 1 C 50 C E∗
c mc> 1 1 / 1 . 7 1C0 F-",G
η∗long − η∗maxlong ≈ 1
2ln
[E∗max
c − p ∗maxc ‖
E∗c − p ∗
c ‖·
2p ∗c ‖
2p ∗maxc ‖
],
E∗maxc − p ∗max
c ‖ ≈ p ∗maxc ‖
(1 +
m2c
2(p ∗maxc ‖ )2
)− p ∗max
c ‖ =
=m2
c
2(p ∗maxc ‖ )2
, F-"*G
E∗c − p ∗
c ‖ ≈ p ∗c ‖
(1 +
m2c
2(p ∗c ‖)
2
)− p ∗
c ‖ =m2
c
2(p ∗c ‖)
2,
1 2 | xF |≈ 1 11 1 p ∗c⊥
2 9 12 p ∗c ‖
0 1/
η∗long − η∗maxlong ≈ ln
(∣∣ p ∗c ‖
p ∗maxc ‖
∣∣) = ln (| xF |) , F- !G
5 1 QBR
xF ≈ exp(η∗long − η∗max
long
), xF > 0 ,
xF ≈ − exp(| η∗long | −η∗max
long
), xF < 0 . F- "G
*A
)(& &
; C C2 C 1 0 / 1 a+b→ c+d / 1 1 0→ 1+2 7 0 / / 7 / 1 0 5 5C 1 ”” (a + b) 5 1C 5 0 M0 =
√s0
C 5 (a+ b)0 B1 p0 = pa + pb F E"G 1 c ≡ 1 d ≡ 2
H E" #0 . a + b → c + d1 4 0 → 1 + 2 1 4
.7 2 1 0 72 1 1 1 1 0? C C C / s M2
0 I C C 1 95 72
*B
1 1 10 0 2 156 C 1p0 I 0 1 1 / / 1
/ '& $ 0"
&1
; 0 1 1 72 T 0 2 156 0 M00 9 2 19 1 1 7 2 ! T0 1 1?
T0 =M0 −m1 −m2 , FE"G
2 9 M0;1 16/ s → M0
0
E∗1 =
M20 +m2
1 −m22
2M0
T ∗1 = E∗
1 −m1 =(M0 −m1)
2 −m22
2M0=
= T0m2
M0+O
(T 20 /M0
)FE G
T ∗2 = E∗
2 −m2 ≈ T0m1
M0.
; 0 72 1/ 0 11 12720 11 / ?
T ∗1
T ∗2
=m2
m1.
<0 1 0→ 1+20 1M00 1 1 1 / 1
p∗1 =
√2m1m2
M0T0 . FEAG
*-
I1 1 ” ” 1 1 2
/ 2
; 0 1 ” ” 0 B1 P0 = (E0 ,p0)F 1 Z 1 15 1 p0G . 1 1 1 B1 P1 = (E1 ,p1) . 2 1 1 2 C p1 1 Z L 2 0 (P0 · P1) 1 156 ?
(P0 · P1) = E0E1 − p0p1 = E0E1 − p0p1 cos θ1 = E∗0E
∗1 . FEBG
. 72 1 156 0 72 1 1 0 9 FEBG C 0 56 11 2 H9 7 ?
p1 =M0E
∗1p0 cos θ1 ± E0
√D1
E20 − p20 cos2 θ1
, FE-G
2D1 =M2
0 p∗12 −m2
1p20 sin
2 θ1 . FEEG
H9 FE-G 6 0
sin θ1 ≤ M0
m1
p∗1p0
. FE+G
. 1 7 0 ”1 ” /
" M0
m1
p∗1
p0> 1 I2 C 5 2 F 0
180G0 2 2
M0
m1
p∗1
p0≤ 1 7 0 0 2 2
M0
m1
p∗1
p0= 10 C 9 *!0
C M0
m1
p∗1
p0< 10 C F*%G
2 ?
θ1 , = arcsin
(M0
m1
p∗1p0
)= arcsin
(γ∗1β
∗1
γ0β0
)FE,G
*E
. 7 F 1 12 A -G
" 1 ”2”0 1 N F 5 2 G0 2 2 F 2",!0 G I 0 1 2 C 10 1 1C 15 1 156
1 1/ 0 56 5 0 1 5 9 CM0 m1 0 0 5 F0 0 TG 72T00 2 F9G 1 p0 156 0 1 1 12 7 2 C0 1
p0 ≥ p∗1 ·M0
m1, γ0β0 > γ∗1β
∗1 . FE*G
; 0 1 9 1 156 1 1 5 ”1” @ 2 $1 Q R?”. 7 2 N 1/ 156 5 10 1 156 C 1 ”” 1 1560 1 ”1” 0 1 1 2 9 0 C ”1”” . 7 FE*G 0 2 1 10 9 1 p00 1 1 1 2
A C 1 0 56 5 0 1 5 9 C M0 m1 0 05 1 p00 2 72? 0 1
p∗1 < p0 · m1
M0, FE"!G
1 1 1 2
*+
/ 2
H 1 25 0 N 1 2/ 1/ C / 1 0 → 1 + 201 1 2 ψ(E1) 2 C 1 p1 p2
1 72 1 1 E1 H 7 1 1C 71 1 1/ F E G
H E ! 0 ! 1 4+ ' ! ! ! 0 J(L
81 1 0 7 56?
(P1 + P2)2= P2
0 ; q2 =1
2
(M2
0 −m21 −m2
2
). FE""G
; 12 FE""G 2 1 0 1
*,
0
cosψ =E1E2 − q2
p1p2. FE" G
8 1 E2 p2 E1O 1 // 1?
cosψ =E1 (E0 − E1)− q2√
E21 −m2
1
√(E0 − E1)
2 −m22
, FE"AG
1 1 E10 0 5 1 1 0 ”” FE1 ,G ”1” FE1 ,G 0
E1 ,min/max =E0E
∗1 ∓ p0p∗1M0
, E1 ,min ≤ E1 ≤ E1 ,max . FE"BG
%9 1 1 1/ FE""G 0
E1E2 =M2
0
2 (1− cosψ), FE"-G
C 0 2 C 50 72 156 725?
E1 =E0
2±√E2
0
4− M2
0
2 (1− cosψ), FE"EG
E2 =E0
2∓√E2
0
4− M2
0
2 (1− cosψ), FE"+G
0 9 2 20 2
sin
(ψmin
2
)=M0
E0=
1
γ0FE",G
cos
(ψmin
2
)= β0 , FE"*G
**
1 7 65 12 C FE"EG FE"+G?
E20
4=
M20
2 (1− cosψ). FE !G
9 FE"EGFE !G?
" 72 156 2 1/ 9 O
2 2 C 725 156 F G F 72G
"!!
%
*+& ,
3 %
4)!"&
$2 7 / 6 1 1 C N 5 N 2 2 / 6 > 2 6 91 0 0 1 2 7 / 2 C 20 0 0 2 1 1 71
; 0 ”2 / ” 1 5 8 1 0 1 / / ? K0
S Λ0210 156/ 16 1 C/
"!"
H +" 8 . Λ0 Λ0 K0S 0
-M J,PL !.!! 3 =HQRBFF V1 ' 0 1 4 V2 ' 0 1 V 004+ . Λ0 Λ0 & ”” !". Λ0 Λ0 K0
S 0 M . . 0 3! ! . 0. ! V 0 ” ” 1!4 " V1 V2 ! θ ! 3 . . ! ! 0
71 / 1 5 / C 1 K0
S Λ0 /0 2 1 1 10 7 1 1 F10 0 G I12 1 7 / 1 ? 1 1 1 1C C ? 1 1 K0
S 7 N 1C C π0 1 1 Λ0 N 1 C 1 8 / / 1 2 0 5 125 1 V 0 ””0 19 N V 0
"!
9 7 0 2 1 0 56/ V 0 ””0 . Q-ER 1C "*-B 2 6 6
. 7 2 1 V 0 / 112 F1 95 15 C V 0 G 1 p t 1
α =p+L − p−Lp+L + p−L
,
/ 56 5 C 1 F1 95 C 15G 1 1C Fp+L G Fp−L G C/ 7 V 0 ”” I C 5 1 V 0 C 1 5 1 (α , p t)
I1 1 71 1 . / 1 V0 1 1 1 1 1 0 7 2 1 1 900 1 1/ 5 11 1 1/ 1 1/ 1 C V01 0 1 1 2 0 180 10 1 1C 1 1 1 56 71 1O 7 71 1 71 10 1 5 F1 1G 0 0 1 V0 C 1 F 1G0 56 71 .7 1/ 1 1/ 0 112 1 F172 1 1G 0 2 V0 Λ 21 0 1 1 (α− pt) C 2 1 V0
F Q-ER +"G? 1 (α , p t) K0
S 1 1 5 5 9 25 0 1 Λ00 1 F G ; 1 7 16 5 F10 G
"!A
3 +#" ” "”
1 C 1 C56 /6/ 7 / 0 15 N
. 7 ? / 0 C Λ21 2 210∆ C / Q-,R 5 1 / F1C5 2 1 29G ∆0 C2 1 7 / 1 8 1 95 2 1 / 1
8 0 1 2 71 C 1 F 1 0 C0 G0 0 C / C C 9 F 5 5”C”G ; 0 / 1 0 0 C C 1 1 F 7 G > ” ”
C 1 ” ” 1 .2 QE!R 1 1 21 / 8 0 K + p → π + Λ0 2 1 1 0 11 -A! 7U Λ21 C 16 F 1 9G > / 1 21 165 (K,π)0 1 ”2 1” 7
=5 10 1 C 1 ” ” 0 2 0 71 1 2 15 71 10 5 L && . 7 2 1 / /C / 1 10 1 / 15
0 125 5 1 b + A →c + A′ + d0 2 bN 1 FG0 AN9 FG0 cN 56 0 dN 9 2 0A′ 0 1 56 1
"!B
F1 G 1 1 0 1 ” ” c0 C 1 A′ 7 7 C c / FG 2 d0 /6 9 7 C 0 1 1 c C 1 1
$ 0 ” ” 0 c 1 50 9 FAA,G
pc =(pc⊥, pc ‖
)=
= (0, γcmE∗c (βcm − β∗
c )) = (0, 0) βcm = β∗c ; F+"G
0 0 0 7 2 0 . 7 C ””
H 71 1 c 1 1 5 75 1 0 12 0 12 1 0 2 0 1”1” 71 12 pc ‖ ? 62 71 F 1 AEG 1 1 5 9 1 ?1 710 2 1 0 6 20 2 1 pc ‖ FAAEG FAA+G > 71 1 1/ 12 0 1 C A-
c 0 β∗
c 2 C 1 0 71 1 C ” ” 2 1 I 1/ 0 10 1 p+p→ pp+jbk`l0 C mc > mp
C c 0 ” ”C 10 2 1 71 1 c p⊥ 1 1 710 C/ / A-
"!-
, GeV/cbeam
p0 2 4 6 8 10
, GeV
/cm
inq
0
0.1
0.2
0.3
0.4
0.5
0.6
H + 6 ! 1qmin4 ! 0 1Λ 1! 4 η 1 4 ω 1 4 4 !! ! !0 1pbeam4
, GeV/cbeam
p0 2 4 6 8 10 12 14
, GeV
/cm
inq
0
0.2
0.4
0.6
0.8
1
, GeV/cbeam
p0 1 2 3 4 5 6
, GeV
/cm
inq
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
H +A 6 ! ! 1η ω η′ 100 4 φ 1 44 !! ! !0 1! .0 0 S(4
H +B 6 ! ! 1 0 S(4 !! ! !0 0 0
"!E
80 9 c0 C/ 1 1 10 56 5 ” ” 0 1 1 A′O 7 / 0 ”7 ” 2 C C 55 1 2 180 8 C/ 9 1
C 1 c FqminG0 2 9 F+"G?
qmin ≡| pc ‖ |= γcmE∗c | (βcm − β∗
c ) | . F+ G
+ qmin 1 1 p (K,Λ)π F9 /1 G0 p (π, η)p F 19G p (p, ω)pp FC 19 G 0 C0 C 0 9 1 10 56 15 ” ”
+A 2 2 p (π, η)p 0p (p, ω)pp0 p (p, φ)pp p (p, η′)pp 0 pp / Fη′ φG0 / 9 1 0 /C 1 C
. 1 1 1 / 1 1 90 9 C C/ 2 C ” 2” 7 C 0 1 9/ s0 10 9 2 F 2G0 0 C/ 15 ”2”1 F +BG > 1 / 2 F 1 AE0 2 92 71G C0 1 C C C ” ” C 1 Fη′ φG1 / / 1/
; +-0 +E0 ++ C 0 5 2 1 1 1 1 /”7 / ”0 / / 2 C 0 9 / ” N ” F 1 1 AE0 7 N 2 92 71G
"!+
, GeV/cbeam
p0 2 4 6 8 10 12 14
, GeV
/cm
inq
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
H +- 6 ! 0 S* !! . ! !0 2 ”3 ” 1.25 × mp mp
, GeV/cbeam
p0 2 4 6 8 10 12 14
, GeV
/cm
inq
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
, GeV/cbeam
p0 2 4 6 8 10 12 14
, GeV
/cm
inq
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
H +E T 0 !" ! ”3 ” 1.30 ×mp
H ++ T 0 !" ! ”3 ” 1.35 ×mp
"!,
- % ; 6 %
L && /C 1 72 c0 C b + t → c + X O b N 0 t N 9 mtarg0 16 0 X N 5 M F 56 G?
E∗c =
s+m2c −M2
2√s
, p∗c =λ1/2
(s,m2
c ,M2)
2√s
.
I C 1 0 1 / 72 1C
βcm ≈ 1− 2mtarg
s.
10 β∗c = p∗c/E
∗c 1 / 72
7 0 0
β∗c − βcm ≈
2
s× (m2
targ −m2c
).
> C 11 2 ?
β∗c − βcm ≈
2mtarg · (mtarg +mc)
s×(1− mc
mtarg
).
I 0 2 9 90 0 C0 120 C 1 ” ” 2 0 1 / 72/ 1C 0 1 1 1/ ; 1 7 1 ” ”
3 )
$
I ” 2 2” 1 7 0 10 / 56 1 122 0 2 0 / 6 20
"!*
5 . 1122 1 ”1” F 2 0G
H +, # (MX − t) (d, d ′) ! 0 U (Mmiss, t) ! 3 Q 0 ! ! # K ( V * ! !0 VSV N3&W **5, N3&W ,,V( N3&W 5) N3&W T0 K))23& 0 Q J,SL
. 2 F 1 N”1”G 12 B1 t ” 56 ”MX 15 C ? 1 72Q = E0 − Ed
′ F E0 72 10 Ed′ 72
2 G?
Q = E0 − Ed′ O t = Q2 − (p0 − pd′)2 O
M2X = (Q+Mtarg)
2 − (p0 − pd′)2 . F+AG
> 0 15 7250 71 t0 ” 56” MX F +,G
3 % $$ &&
$ C 1 L &&0 2 12 5 ” 7 ”
""!
F ” 7 ”0 10 1 1 /G0 2 122 C F 9G 9 $ 9 F10 G 1 % 20 1 9 9
; 0 12/ 12/ 0 16 FG 90 C6 2 C % 20 C 0 9 m2
bound = E2 − p2 5 56 56 5 O /C / 1? / 7 7 /0 / Q*BR C∆ / F Q-,RG
$ 5 2 / 1C 1 6 0 1 +*
H +* 9." 0 !! *! Pproj ! AX *! Ptar ? 0 !*! Pdetect . ! . *! Pejected*! ! 1 . 4 (A− 1) .0 Precoil 0! *! PN
"""
. 12 7 A 0 N 7 0 C 7 0 (A − 1)N F6 20 CG
. C ∆ / 1 A(p, n) A(3He, t) A A 0 0 10 ∆ 0 (A−1) B1 Precoil F1 7 C0 7 /0 C 6 / / 2 G
. 0 2 9 1 F G0 Ptar = (MA , 0)
! 3 3 Q*BR 1 1 F1 G 1 / 1/ 7
. 1 1 / 565 1 1 15 5 0 1 C 1 11 5 5 1 8 7 561 2 . 1 1 0 115 15 56 1 ? 1 0 0 / 0 /5 1 C 1 1 1 2 / 2 2 56 1 0 56 1 0 10 1 1 2 .7 1 C 2 112 2 / C / 0 C 1 ” ” 2 C 2 9 0 C ; 7 1 16 Q*BR
! ! 6 ∆0 % 0 7 PN B1 2 0Pejected N B1 C ∆ . 1 F9Z ∆G /1 Ptransf = (ν , ∆p)0 2 ν N 1 72 F 7209 1 12 G0 ∆p N 1
""
A1 F / GL /1 0 2 A0 PN =
Ptar − PrecoilO 2 1
PN = (EN , pf ) = (Mtar − Erecoil , −pf ) =
= (Mtar −Mrecoil − Trecoil , −pf ) , F+BG
2 pf N / 1 C 9 A0 Mrecoil N 1 (A − 1) 0 Trecoil N 728 9 C 1 5 1 72 F G 17 C C /9 5 0
Trecoil ≈p2f
2Mrecoil. F+-G
(A − 1) 0 7 C 0 ! εs?
−εs =Mtar − (Mrecoil +mN ) , mN − εs =Mtar −Mrecoil , F+EG
2 mN 2 6 20 C (A−1) 2 C/ 0 / /2 2 71 0 F+EG 7 5
0 F+EGC 1 1 1 / O 7 F+EG 1 εs 1 ! 0 Mrecoil 5
0 m∗N 2 0
19 ! 1 516 Fbab:cbmG F N G1 2 B1 PN
m∗ 2N = P2
N = (mN − εs)2 − Mtar
Mtar−mN+εs
· p2f , F++G
""A
2 O 7 1 5
P2N −m2
N = −2mNεs − Mtar
Mtar−mN+εs
· p2f + ε2s , F+,G
/ ” / 1/ ” 7 2 16 F G 0 7 1 11 720 / 20 2 ; 12 C0 2 122 ” ” 2 C 2 F2 C 2 0 1 1 1 εsG
16 F G Mejected 1 P2
ejected0 0
P2ejected ≡ ω2
F = (PN + Ptransf )2= (ν + EN , ∆p+ pf )
2, F+*G
C 1
ω2F = t+ (mN − εs)2 + 2ν (mN − εs)−
− (Mtar + ν) · p2f
Mtar −mN + εs− 2pf ·∆p . F+"!G
@ t N 1 F 12 B1G ; 12 9 0 6 2 ? 2 C 1 A1 A1 2 C 0 19
122 FG0 ωF 0 0 8 0 2 7 N F1∆G0 6
1 / 2 22 RO 7 5 ”26” ω12 2 ” ” ω00 C C 0 56 15 9 ΓR 1 (1 + 2)?
σ ∼ ΓR (R→ 1 + 2)
(ω20 − ω2
12)2+ ω2
0Γ2R/4
. F+""G
""B
”%26 ” ω12 7 (1 + 2)0 1 7 / / / / 1/ / . 9 0 ΓR (R→ 1 + 2)0 C 0
6 0 (q/q0)2l+1
8 C 1 2 / C2 1 R→ (1+2) @ q N 1 1 2 / 1 7 (1 + 2) ω120 q0 N 1 ω0
2 C 0 /6 56 (1 + 2)0 / ' %' ' ? B1 PN 0 B1 Ptransf 0 1 7 F +*G .7 1 0 1 C ω120 5 1 2 / / 1/ /0 ωF 0 2 7 / / /1/ /D F./C 1 C C Q-*RG
8 0 1 ω12 = ωF C 2 56 2 ?
" ωF 0 ”1” q∗ 21 1 F
2 0 1G 1 20 1 0 2 1 B1 Ptransf ”” P2
transf O
C C 0 q∗ 21
1C 1?
q∗ 21 =
λ(ω2F , m
∗ 2N , t
)4ω2
F
; F+" G
√s120 15 725
C F G 2 F /6 1/ G 1 1q∗1 7 O
A C 7 ”26” 0 1 ω12 =
√s120 1 E∗ 2
1, r = q∗ 21 +
m2N ≥ m2
N
E∗ 21, r 1 72
1 7 2 F
""-
G O 10 1 C∆ 1 2
E∗ 21, r =
(ω2F − t+m∗ 2
N
)24ω2
F
+(m2
N −m∗ 2N
), F+"AG
2 m∗ 2N N 2 F F++GG
""E
- .
/
8 71 7 / H / 2 2 ; / / 0 15 0 / / / 2/ 10 1/6/ 1 7 / > 0 5 0 1 ”112 ” 0 2 1 1 ”” ” 1 ”0 2 1 3 7 15 / 7210 9 2 10 ”” Q RF 0 11 0 ” ”0 ” ”G / 1 1 2/ 0 / / / F10 0 1G
, " + / $( 0( $ " "
"",
= %>9 7 C 1
. C 7 / 1/ 10 1 1 0 11 7 71 7 2 1 1/ F72 / 1 L &G 0 1/ / 1/ /0 C/
5 .&
! 1+/& 4(. +%44..
1 1 112 . 5 1 1 na0 nb va0vb F G 1 b
a v(b)relO 1 vrel : " dν
K dV dt 0 0
dν = nb dV na · vrel · dt · σ , F,"G
2 σ N ; ? vreldt na · vrel · dt
1 a .60 1 7 9 b0 nb · σ @ 7 11 σ 2 1 1 / / F aG 5 9 F bG? a0 119 1 5 9 0 1 1 5 0 1 F,"G C !0 1 0 0 7 1 F 9G 22 C 1 / 0 7 11C 1 1 0 2 1 0 10 / 3N
""*
7 2 0 5 @ t ct0 2 c 1 1 20 F,"G1 8 / 1 C 0 7 1 @ 1C 1 0 0 C A A
# 0 C 0 C 0 1 F,"G
dν = n(0)b dV n(0)
a · | uab | ·c dt · σ F, G
1 0 7 1 1 F!G 1 0 1 56/ F11 0 1 / KG6 F, G 0 uab * 0 a b
6 5 20 0 10 / 1 C C H 6 F,"G 165 1 g2 2 1 0 0 1 1 B 0 C 1
InvF lux =
√((Pa · Pb))
2 −m2am
2b = mamb · | uab | . F,AG
F@ Pi N B1 i0 mi N Gg 1 C 1
0 1 sab N 1 72 a b 1 FA *G 2?
InvF lux =1
2· λ1/2 (sab,m2
a,m2b
)= p∗
√sab , F,BG
2 p∗ 1 a b / O 1 12 1 9 FA *G= F,BG 12 90 1 1 71 /0
" !
2 1 C2 56/ / 1 F1 2 0G 1 p∗
! % :+.25 1,1+(9 4(.>
8 7 1 1 ? / F,"G F, G 1 1 0 2 71 2 0 1 9 b
1 I2 n(0)b dV = ρ · l ·dS ·NA/A
9 l 11 dS0
n(0)a c dt | uab | N 10 19/ 16
9 dt 8 11 1 9 9 F 90 10 1 5 1/ 1GO17 C dS C 1 0 65 1 I2 T 9 1 11 S 0 2 / 0
ν = σ · ρ ltarg · NA
Atarg·| uab | ·c
∫dS dt · n0
a
=
= σ · ρ ltarg · NA
Atarg· Iτ· δ ·Ncycl , F,-G
2 Ncycl N T 0 δ N nm\copq:c`_nFC G0 τ N 1 0 I N 1 > C 11 0 1 L
ν = σ · L , L = ρ ltarg · NA
Atarg· Iτ· δ ·Ncycl . F,EG
F,EG 1 ”71 1C 95” F9 1 G 8 7 1 6 1 71 /0 2 5 1 C 1/ 7
9 20 2 2 1 n n1 @ 1 2 / C2 10 2 56/
" "
8 C/ 71 5/ / N 1 0 1 20 1 9? 12 0 12 C 5 56/ 0 12 F C 52 0 1 2 / C2 1 72 G
5 6&
L 1 0 C 1 0 56 1 . 1 / 1 1 1151 6 x 1 15 1 $ P0(x) 20 1 1 6 1 1 2 F10 G 6 D
80 56/ 1 10 16 1 F 56/ G0
Ncenters(x) = x · ρ · NA
M × nmol , F,+G
2 ρN 1 6 1 0M N 6 0 NA N 20 nmol N 0 C6/ 7 0 6 9 . 1 σtot 160 1 56 1 10 Ncenters(x) · σtot ; 1 I00 0 156/ 9 0 ; 10 1992 1 0 1 2 0 0 0 I(x) = I0 · P0(x) 01 56 C
# 6 1 5 dxI0 1 7 2 1 11C 1 2 0 0 0 6 / 56/ 1 / / ?F"G 2 1 1/C x0 F G 2 1 1/C 2 dx 12 2 P0(x) $ C
"
2 2 D 80 0 2 556/ /? 2 F 2 G0 0 2 Ncenters(dx) · σtot 1 7 / "0 0 P0(dx) = 1 − Ncenters(dx) · σtotI 0
P0(x+ dx) = P0(x) ·P0(dx) = P0(x)(1− dx · ρ · NA
M ·nmol · σtot) . F,,G
7 0
dP0(x)
dx= −P0(x) · ρ · NA
M · nmol · σtot . F,*G
86 9 7 2 /9 O 1 2 P0(0) = 1 F 1 6 1 5 " G0 1/ 1 F 2/ G ?
P0(x) = e−x·ρ·NAM ·nmol·σtot = e−x·σtot·ncenters , F,"!G
2 ncentersN 56/ 6 1 0
I(x) = I0 · P0(x) = I0 · exp(−x · σtot · ncenters) . F,""G
= F,""G C 2 9 1 / 1 71 2 1/ 7 1 5 1562 1 10 1992 1 9 I % H 2 C 1 0 //? 1 2 5 0 19/ 9 2 20 2 C 1 0 19 9 C 86 / 1/ 7 R(x) F," G1 ,"
" A
H ," 9.." 0 1 !4 0 3 ! ” ” 1 !4
5 7 " "
7 71 1 N /C ! 1 ; 12 R?
R(x) = I(x)
I0=N(x)
M= exp(−x · σtot · ncenters) , F," G
2N(x) = I(x)·Tmeasur 0M = I0 ·Tmeasur0 Tmeasur N 3 0 6 1
/ F1 1 G ' 129 0 0 ?
" 1 2 5 11 O C 2 5 5 F5G25 / O
1 15 1 F G 2 5O
A 6 9 ”” FC56 G 6 ? 10 6 0 2 56/ F 156 9G O
" B
B 5 "!!r 7 5 2 O 0 F G
- . 1/ 9 C 5 1 2 2 2 2 Ω 2 2 . 7 C 90 C 1 C 0 2 / 1 7 1 2 @ 7 1 C 0 1 7 1 N C 1 7 1 N C 1 1 129 ? 0
; 0 165 / 0 15 ," % 9 ” ”0 5 2 9 " C 1 1 5 2 129 $ 0 7 0 1 71 F," G F C -,-B L sG % 112 0 1 0 2 11 2 / C F 0 11 1 9 / 71 1 5 G 0 1 0 ” ”0 2 i0156 2 Ωi
7 / 11C/0 F C -, L sG0 0 2 / 0 156 2 Ω 1 90 2 2 2 C 1
N(x,Ω) = N(x, 0) +N(x, 0) · x · ncenters ·∫ Ω
0
d σ
dΩ ′ dΩ′ =
= M · e−σtot·x·ncenters
[1+
+ x · ncenters
∫ Ω
0
d σ
dΩ ′ dΩ′]
F,"AG
C 2 Ω F,"AG 0 C 2 FG C 2 1/65 9 0 7
" -
1 ?
N(x,Ω) =M · e−σtot·x·ncenters
[ ∫ θ
0
f(θ ′) sin θ ′dθ ′ +
+x · ncenters
∫ θ
0
∫ 4π
0
f(θ ′′) · d σdΩ ′ (θ
′ − θ ′′) dΩ ′ dΩ ′′]. F,"BG
@ 1 f(θ) 1 1 20 0 1 9 2 2 0 C 2 1 2 θ / 2 (Ω,Ω+dΩ)O ”11 ” 8 C 5 7 2 0 11 2 90 7 / / 2/ 1 1. θ ′′ θ ′ 1 0 1 2 2 θ ′′ 2 2 θ ′ − θ ′′ 2 θ $0 5 1 1 0 11 1 2 / C C 7 0 5 2 10 1 / 2 7 6 F0 11 2 71 / G
= F,"BG 0 1 2 6 8 1 1 C 1 / / 2/ 1 7 1
. C 1 F,"BG? 10 2 20 11 f(θ)0 0 6 2 θ > θ00 2 dσ/dΩ 6 2 / FΩ ,"G0 1 2 F,"BG C " 1 0 1 0 / C 1 2
# $ $ $ $ * ( ( 0 & 1
" E
5 8 $ &)
&
. C C 1 ! 0 9 125 /C 12 2 ? 7 / 20 2 1 1/C 6 12 1 ! ! 8 3 3 ! J 1 2 2 2 0 0
θmoliere =√213.6tbh
βcp· z ·
√x
X0·[1 + 0.038 · ln
(x
X0
)], F,"-G
2 p0 cβ z 1 F 7UG0 0 x X0 N 6 6 2 ; ! θ ≥ (3÷ 4) θmoliere 2 1 9 2 2 0 2 F1 C/ 1 2 G0 N 11 | t |−20 1 1/ F10 1 1 G / t ∼ 10−3 372c−2 1 > " . 9 2 1/ 1 2 $ 3 * ! ! Ω ) ' ; #; ! ! + 6%* 7 % 0 0 ! ! ! % 3 ! ! !
80 2 2 Ω 6 2 / > 0 1 9 1 | t | 5 ” ” 562 K F G0 11 9 11 1 11 1 11 1 ; 7 25 / 0 / 2
" +
2 0 0 0 C ”1/ 2/ ” F / / C F,"AGG > /9 1 122 2/ /F ,BG .7 7 1 ”” 2 9 1/ 2/ F 1 0 0 C / G0 7 C 2 0 / ," 12
H , % !! 1 4 1 4 JS5L
; 0 1 1/ 2 0 ' ; ;” ! ” 2 F , ,BG
8 2 2 F1 | t |G / / F /0 2 7 C 6 G 12 2 / /
I10 10 11 0 C 1 5 2 7
" ,
H ,A % !! " 1 . JY)L4
H ,B / 0 !! Tkin ∼ K N3& M8:Z JYKL / 0 !! * + 98:8 JY(L
" *
/ "& "!0
16 2 C 1 112 11/ / 2 71 / 15 % 1 0 1 1 ” 1 ”0 2 1 8 7
9 .& &&
. / 1/ 1 1 0 S 0 56 F|i >G F|f >G 1 8 1 0 2 7 1 C 12 2 1 20 19 <0 7 S 2 20 2 1 B1 2 Pf 1 B1 2 Pi0 1 / 721 > /F0 0 G
. S 6 /0 17 2 1
"A!
/ 1 C F1 G < f | ← |i > 7 <f |S|i > δ5 / B1 1 2 7 119 2 ” 1/” T ?
< f |S|i >= δf , i + i (2π)4 δ (Pi − Pf ) ·N · < f |T |i > , F*"G
2 N N C 0 0 01 / C / C 0 56/ F 20 2 G 0 5 1 F7 1 1 G0 C
1/ |i > |f > 2 7 ?
δW
δt= (2π)
4δ (Pi − Pf ) · |N |2 · V · | < f |T |i > |2 , F* G
2 V K0 1 1 C F* G 1 9
[δ (P)]2 → δ (P) · 1
(2π)4
∫dx eiPx =
V t
(2π)4 δ (P) , F*AG
2 2 1 K 1
10 16 1 C2 10 0 1 δW/δt 1 156/ 1 1 21/0
σ =V
u\v· (2π)4
∑
|N |2 ×
× δ (P2n , f −m2
n
)θ (En , f ) · δ (Pi − Pf ) · | < f |T |i > |2 , F*BG
2 Pn , f N B1 n 2 |f > F 0Pf =
∑n Pn , f G0 C 0 2
/ 1/ 0 Pn , f = m2n0
1 1 72 n 1C F17 F*BG1 G
"A"
. 156/ F10 1 9G0 1 / ” K”0 1 95 / 2K? u\v = v0/V O 17 F*BG 1
σ =V 2
v0· (2π)4
∑
|N |2 ×
× δ (P2n , f −m2
n
)θ (En , f ) · δ (Pi − Pf ) · | < f |T |i > |2 , F*-G
2 v0 C C 9 17 2 1 ?
|uab| = γav0 =Ea
ma
|pa|Ea
=|pa|ma
mb
mb=
=
√E2
am2b −m2
am2b
mamb=
=
√(PaPb)
2 −m2am
2b
mamb,
0 1 F 1 InvF lux F,AGG
EaEbv0 = mamb|uab| = InvF lux F*EG
8 5 0 C / a F bG
Na =1√
V · Ea
, F*+G
F*-G 1
σ =1
EaEb v0· (2π)4
∑
|N1|2 · δ(P2
n , f −m2n
)θ (En , f ) ·
δ (Pi − Pf) · | < f |T |i > |2 , F*,G
2 N1 N C C 2 8 7 C
• . 1 1 ?
"A
= 1 1 O
= 2 1 B1 2
• . 2 1 B1 7 K d4Pn , f O 0 20 56 / 1/ F 0 P2
n , f = m2nG 72 1C 0
2 1 211/ 0 .7 / 1/ 1 1 5 7 2 0 1 2 1 72F F*"-GG
• I1 1 0 1 2 C F*+G 1 72
20 C F*,G C 11
σ =1
EaEb v0· (2π)4
∑
∫...
∫dq1
2E1...dqn
2En|N1|2 · ·
δ
⎛⎝Pa + Pb −
∑j=1 , n
Pj
⎞⎠ · | < f |T |i > |2 , F**G
; / 0 7 K d3p
1 1 5 (V/ (2π)
3)d3p 0 1
C 1 .7
σ =1
EaEb v0· (2π)4 ·
(V
(2π)3
)n∑
∫...
∫dq1
2E1...dqn
2En|N1|2 · ·
δ
⎛⎝Pa + Pb −
∑j=1 , n
Pj
⎞⎠ · | < f |T |i > |2 , F*"!G
= 0 / C F 5 C 1 (2π)3G 0 K F*"!G 6 56 01 |N1|2 2 1/ 0 2 2
"AA
K0 2 C 0 C ?
σ =1
EaEb v0· (2π)4 · 1
(2π)3n
∑
∫...
∫dq1
2E1...dqn
2En· ·
δ
⎛⎝Pa + Pb −
∑j=1 , n
Pj
⎞⎠ · | < f |T |i > |2 . F*""G
C 2 29 / / ? 7 2 9 C0 562 7 0 8 F*""G
9 !& :
3 n 0 / B1. 0 0 1 1 > K B12 1 n F2 C 5 7 G ?
dRn = d4P1d4P2....d
4Pn F*" G
1 K 7 2 2 1 C ! " 8 0 1 F"G 1 72 n OF G B1 C 5 ””0 P2 = E2
i − p2i = m2
i 0 F2 0 7 / ” 1/ ”G L 7 0 7 2 K 1
dRn =
n∏1
d4Piδ(P2
i −m2i
)δ4
(n∑1
Pi − Pn
), F*"AG
2 Pn N 1 B1 9 n 1 9
2 1 1 1 1/ 2 56 FM2G? C 2 7 2 K dRn0 1
"AB
dW 20 56 ?
dW =M2 ·n∏1
d4Piδ(P2
i −m2i
)δ4
(n∑1
Pi − Pn
), F*"BG
1 2 1 72 1/ F/ 15G / 0 C 1 15 1 1 562 10 7 16 F F*""GG
C 725 E1 F 1G 0 10 10 1 2 1 10 dRn C 1 F (E1)dE10 2F (E1) ”16” ”” 2 1 1 5 E1 = constO 7 1 1 725 E1 .6 20 7 0 72 1 1
H F21 0 / 1G 0 2 ! 3 0 * 0 0 2 20 C 2 1 0 1C 7 1 B ! * * % 0% ' % * . ! 7 ! 3 ' 0! &* ; 10 1 1 11 1 K0 1 U1 1 / // I 0 7 1 5 2 K0 N Q RT
; 0 1 M2 ≡ 1 F 0 5 C1C "0 5 GI2 F*"AG 1 7 2 K > B 0 6 B10 1 0 2 20 % 20 1 2 1 Pi C 12B1 / FG0 1 2 2 > 0 1 K C 1 F 1 sG
"A-
@ 0 / 1 C2 Pi A 5 F / 1/ G0 C δ F*"AG0 1 2 1 1 ?∫
d4Pi δ(P2
i −m2i
)=
∫d3pidEi δ
(E2
i − p2i −m2
i
)=
∫d3pi2Ei
, F*"-G
2 1/ 1 7 1 δ
0 7 2 K F*"AG 1 FC 1 A1G
dRn =n∏
i=1
d3pi2Ei
δ(3)
(p−
n∑i=1
pi
)δ
(E −
n∑i=1
Ei
). F*"EG
C F C 2 %52 $ QARG?
" F*"EG 1 C C 0 7 % *% 3! !' δ0 % 0% 6 ) 7 C+ O
7 2 K0 1 F*"EG0
; 0 K n
Rn(s) =
∫ n∏i=1
d3pi2Ei
δ(4)
(P −
n∑i=1
Pi
). F*"+G
2 / F Q RG0 10 0 / ?
R2 =
∫d3p12E1
d3p22E2
δ(3) (p1 + p2) δ(E1 + E2 −
√s). F*",G
. 2 1 p2 δ0 1?
R2 =
∫d3p12E1
1
2√p21 +m2
2
δ
(E1 +
√p21 +m2
2 −√s
). F*"*G
2 $ 34 & 2π
"AE
1 ? d3p1 =p21dp1dφd cos θ0 2 F10 1 " TG 1 p1 0 2 φ 0 2π0 cos θ −1 +1 2 1 1 4π
; 2 1 p1 C 2 1 2 δ? 0
p21dp1 = p1E1dE1 ,
E2 =√E2
1 −m21 +m2
2 ,
x = E1 +√E2
1 −m21 +m2
2 −√s ;
dx = dE1 +E1dE1√
E21 −m2
1 +m22
= dE1
(1 +
E1√E2
1 −m21 +m2
2
)=
= dE1
√s√
E21 −m2
1 +m22
, F* !G
dE1 =
√E2
1 −m21 +m2
2√s
· dx ,
0
p21dp11
2E1
1
2√p21 +m2
2
δ
(E1 +
√p21 +m2
2 −√s
)=
= p1E11
2E1
1
2√E2
1 −m21 +m2
2
√E2
1 −m21 +m2
2√s
· dxδ (x) =
= p11
4√s· dxδ (x) , F* "G
2 p1 N 1 " O 0 7
p1 ≡ pc.m. =
√s− (m1 +m2)2
√s− (m1 −m2)2
2√s
,
E1 =s+m2
1 −m22
2√s
. F* G
"A+
I1 2 1 dx 1 O 4π 2 1 20 10 /
R2(s) =πp∗1√s=πλ1/2
(s,m2
1,m22
)2s
, F* AG
2 1 p∗1O 1 C λ(a, b, c) = a2 + b2 + c2 − 2ab− 2ac− 2bc
12 0 2√s
2 0 K F 0 2
√s 12 5G
R2(s) ∼√√
s− (m1 +m2) =√ε , F* BG
F1 1 ε G0 1 F2 C 1G
Rur2 (s) =
π
2. F* -G
/ ?
R3(s) =π2
4s
∫ (√s−m1)
2
(m2+m3)2
ds2s2
λ1/2(s2, s,m
21
)λ1/2
(s2,m
22,m
23
); F* EG
1 Fmi → 0G
Rur3 (s) =
π2
8s . F* +G
@ 0 7 1
Rur3 (s)
Rur2 (s)
=π
4s . F* ,G
8 0 6 n 1
Rurn (s) =
(π/2)n−1
(n− 1)! (n− 2)!· sn−2 . F* *G
1 1 1 C 2 K /
"A,
0 7 1 1 C n ! sthresh02 s ≡ M2
n sthresh . 7 1 1 ε ≡ √s − √sthresh? ε %& ”%8 ! ”
6 n 1 C 1 1C F1 2 3;$1GO 56 C
Rnrn (ε) ≈
(2π3
)(n−1)/2
2Γ32 (n− 1)
· (∏mi)1/2
(∑mi)
3/2· ε(3n−5)/2 . F*A!G
/ 1 2 ?
Rnr3 (ε) =
π3
2
(m1m2m3)1/2
(m1 +m2 +m3)3/2·(√s−
3∑i=1
mi
)2
=
=π3
2
(m1m2m3)1/2
(m1 +m2 +m3)3/2· ε2 . F*A"G
0 1 p+ p→ p+ p+ V 0 2 V 0
Rnr3 (ε) =
π3
2
mp
(2mp +mV )·(
mV
2mp +mV
)1/2
· ε2 . F*A G
2 0 1 C / V1 V2 / 120 9 / K F1 19 56 12TG
Rnr3 (ε;V1)
Rnr3 (ε;V2)
=
(mV 1
mV 2
)1/2
·(2mp +mV 2
2mp +mV 1
)3/2
. F*AAG
: 8 %' . ;
; F*"+G "!" C 5 0 K n C K 9 0 9 6 5 F 2 Q 0 ARG 8 C0 C C / 2 2K
"A*
$% 3;
H 5 C η 1 1 / / 120 9 1 0 1 F/ 1 7 G 72 5 12 7 F1 N 12G
H *" / 0 .
ppη √s = 2433.8 23& 1 .
3 ! 0 ppη 4 M .7 " ! 0 3
! Vps = π2
4s
∫ ∫dm2
p1ηdm2p2η M!
!0 . η .. ( 23& ” 0” 0 ! JP,L
7 1 C 0 1 N
0 1 0 0 9 0 1 1 0 156 / C . 0 F G 5 1 720 0 9 / 5 7250 20 / 1 ?1 2
1 1 C0 2 // 2 1 0 1 56/ 0 ” ”
"B!
H * &0 . ppη ε = 16 23& S 0 M" 0! . !". @. ! (P) . JP,L
0 72 9 > 7 ”5” 12 2 K F 7 / 1/G
1 0 C 9 2 1 0 1 0 1 1 5 5 725 I 0 C 9 F 7 92 2 KG 1 5 0 1 56/
> C 5 5 *"O * 10 11 ” ” 2K
* ( " $ $ .0
"B"
)& )
12 "
12/ / / 1 9 /0 5 5 / // C /1 F 1 / 1/G %9 C C / 1/0 1/ 1 1 / 2 1 1 5 0
; 7 &&
# %41%/ 1→ 3
H1 P → 1 + 2 + 30 2 N 10 1 2 9 1 0 A 2 F* 1/G0 / 721 I
"BA
H "!" #0 0 *! P 1X *! P1 P2 P3 4 8! s1 s2 1 3 (1+ 2) (2+ 3) 3 3 !" 0 3 4
N 10 1 1 P C / 2 FA / 1/G0 15 5 2 .7 5 9− 4− 3 = 210 C ;/ C C 0 11 0 1
4% % 45 1 12 F"!"G 1 C 21 1 P → 1 + 2 + 3 F "!"O s 1 G?
s12 ≡ s1 = (P1 + P2)2 = (P − P3)
2
s23 ≡ s2 = (P2 + P3)2 = (P − P1)
2 F"!"G
s31 ≡ s3 = (P3 + P1)2= (P − P2)
2
> ?
s1 + s2 + s3 = s+m21 +m2
2 +m23 . F"! G
# %8+%9 %&%
P → 1 + 2 + 3 2 1 Fs1, s2GC 1 0 7 /
"BB
1/ 9 / 721 F 5 5 G
6 1 2 15 5 5/ / 1/0 s1 s2 F 1/ 56 1 1 G > 2 0 10 5 1 Fsi, sjGO5 1 FE∗
i , E∗j GO 5 1 / 72 FTi, TjG0
Fi, j = 1, 2, 3G#0 156 2 2 F 2
1/ P → 1+2+3G0 C ”15” QAR
H K
R3(s) = F"!AG
=
∫ 3∏i=1
d3pi2Ei
δ3 (p− p1 − p2 − p3) δ(√s− E1 − E2 − E3
).
. 2 δ . K 0 C 1 1/ C
1 156 1 2 1 p2?
R3(s) =
∫d3p1 d
3p38E1E2E3
δ(√s− E1 − E2 − E3
), F"!BG
2 F10 0 0 56 1 156 0 0 C 0 F"Z ZAGG?
E22 =| p1 + p3 |2 +m2
2 = p21 + p23 + 2p1 p3 cos θ13 +m22 . F"!-G
1 p1 p3 1 0 0 19 d3p1 d
3p3 ?
d3p1 d3p3 = p21dp1 dΩ1 p
23dp3 dΩ3 =
= p1E1dE1dΩ1p3E3dE3d cos θ13dϕ3 , F"!EG
2 2 Ω3 = (cos θ13, ϕ3) 15 5 1 3 p10 2 Ω1 N 5 1 1 0 1 1 2 1 Ω1 ϕ3 F 1 1CG
"B-
2 1 cos θ13 1 δ5 720 dE2/d cos θ13 = p1 p3/E2 ?
R3(s) =1
8
∫dE1dE3dΩ1dϕ3 Θ
(1− cos2 θ13
). F"!+G
@ Θ N 2 cos θ13 2 @ cos θ13 = ±1 5 2 1 (E1 , E3)0 0 2 2
I1 C 2 2 C 1 F"!-G? (√
s− E1 − E3
)2= E2
1 −m21 + E2
3 −m23 ±
± 2[(E2
1 −m21
) (E2
3 −m23
)]1/2+m2
2 , F"!,G
0 C 0(√s− E1 − E3
)2=| p1 ± p3 |2 +m2
2 . F"!*G
C 1 F"!,G0 0 1 7 1 / / 1O ?
4(E2
1 −m21
) (E2
3 −m23
)=
=[s+ 2E1E3 − 2
√s (E1 + E3) +m2
1 −m22 +m2
3
]2. F"!"!G
. E1 E3 s1 s2 F 7 0 1 F"!"GG < 1/ ∂ (E1, E2) /∂ (s1, s2) 1/4s 1 1 F"!+G 1 s1 s2 1 2 1 2dΩ1 F 1 TG0 1 4π0 C 0 ϕ3 2 9 6 1 2 1 F 0 2 1 7 1 2 9 2πG0 1 C5
R3(s) =π2
4s
∫ds1 ds2Θ
(−G(s1, s2, s,m21,m
22,m
23)), F"!""G
2 G(s1, s2, s,m21,m
22,m
23) N 0 6
/ 1 1 2 2
"BE
C 1 C 2 F"!+G0 1
R3(s) = π2
∫dE1 dE3Θ
(1− cos2 θ13
). F"!" G
; F"!""G F"!" G 0 2 * / 1/ Fs1, s2G0 0 1 C 0” .” ?
d2R3
dE1dE3= π2 ;
d2R3
ds1ds2=π2
4s, F"!"AG
s ; 0 1 1 F2 C 1 C 2 0 1C 2 1 72 1/ 7 G 0 56 2 7 1 1 F 72G 1/ 0 2 F G > 68' 2
0 F"!,G C 6 1 9? 2 2
λ(p21 , p
22 , p
23
)= 0 . F"!"BG
# 1%&59 4&0(% /%8+% %&%
% %;
H 1 P → 1 + 2 + 3 1 156 % / 72 ? Ti = Ei −m 156 M0 ≡ √s 80 1
T1 + T2 + T3 =√s− 3m = Q , F"!"-G
2 Q N 72 1 1 9 F"!"-G 1 1 92 2 1 056 / 2? 5 2 2 F T1, T2, T3G
"B+
2 2 F QG > 1 1 (T1, T2) 2 1 25 0 5 2 F "! G @ 0 / 0 / / 1/ s0 t u0 C C1 25 0 7 1 5 95 FAAG0 2 F"!"-G
H "! 9 ! % 0 3 1 ! 4 % !0/ 1[4 0 1 !4 1\4 0 */K/K 1 !4 1X4 S/*/K 1 !4 T ! η → 3π η′ → ηππ pp → η′ηπ JS)L
C 2 1 / 72 /C 2 ;?
" 1 2F T1 = T2 = T3 = Q/3GO
2 1 (r, ϕ)
"B,
C 0 ?
T1 =Q
3(1 + r cosϕ) ,
T2 =Q
3
[1 + r cos
(ϕ+
2π
3
)], F"!"EG
T3 =Q
3
[1 + r cos
(ϕ− 2π
3
)].
. F"!,G 22 1?
(1 + x) r2 + xr3 cos 3ϕ = 1 , x =2ε
(2− ε)2 , ε =Q√s. F"!"+G
H "!A % % !0 M 0 !" ! ! 9.0 δm2
ij 0 3 m2ij
0 (i+ j) ! 0 (mi +mj)2
?/ η → 3π / pp → 3γ pp → η′ηγ 1. JS)L4
ε N ! N 1 5 / 50 565 2 2
"B*
/ C C 1 2 2 0 1 F "! 0 C 2G
. 2 1/ s10 s2 "!A "!B F 7 (1+2) 0 s10 (2+3) C 1 95 s2G
H "!B % % . . 0! ! 0 0 ! JVL
# ,B80+% 10&54,3 % /%8+%
%&%
# F"!,G 2 2 1 F1 95 / 721G 1 // 1 2 H 2 0 1 156 F 1 1G
0 2 2 1 p10 p20 p3 0 / 1? / 1 2 1 1 H "!- 12 1 0 5 2 1 / 1 C 2 2
I / / 2 2 5 7 / m2
12 = (P1 + P2)2 =
(m1 + m2)20 m2
23 = (P2 + P3)2 = (m2 +m3)
2 m213 = (P1 + P3)
2 =
"-!
(m1 +m3)2O "!- A10 A20 A3 #
7 0 1 0 / 1 15 0 1 p1/p2 = m1/m2
2 / 7 2 5 m2
12 = (√s −m3)
2G O B10 B20 B3 C 7 / / 1 1 56/ 1 1 1C0 5 5 725
H "!- # ! ! 0! % 1! ." 4 T0 B1 B2 B3 0 0 3 m2
12 m223 m2
13+ 0 A1 A2 A3 ' 0 3 0 JVL
; * *
#C 2 0 7 / 1/0 1 2 F0 60 1 2 G 1 F F"!"AGG 8 2 1 1 0 C
"-"
/ F G . C C 11 .0 1 C / / 1 2 / L h&&0 1 1 1 2 1 1 21 1
7 5 1 2 71 / / 1 2 9/ 1 1
H "!E % % !
/ pp → K±KSπ∓ 2/ /
3 =]^_`[a b[]]ca / ! % F 1! ! ! 4 ! 1. pp4 &0 (K±KSπ
∓) 3,3S1 1. JS)L4
71 C2 1 2 1 / 0 1 22 1 0 "!E0 / / (p p) 0 256
; 6 2 2 1 5 10 1 2 1 / / 1/0 "!+ "!,
"-
H "!+ ? 3 1 !" %4 0 / π+π−π0 ! & / pp → K+K−π0+ / pp → K±KSπ
∓ ;< dR ;R 1. JS)L4
"-A
H "!, % % p p → π+ π− π0 / 1[4 + 1\4 . !+ 1X4 Z 0 ! 9 . . !0 ' KW( 1. JPYL4
"-B
H "!, "!+O 0 2 10 1 2 0 1 9 5 (p p) 0 56 / 1 2 1 1
; * <$ $
@ 2 Q R F AA G? ”$2 5 10 2 $2 5 1 / 0 125 2L4 J”
H "!* #0 2 → 3 9.00 + ! t1 t2 1 *! 4
/ 1 a + b → 1 + 2 + 3 10 1 1562 1 pa = −pb 86 1/0 / 56/ 0 -O 1/ / 1/ B F "!*G .7 1 1 2 1 /0 90 1/ 2 1 1 K0 / /
! + $ 56 ,7 $ &% 0 ( "
"--
8 0 1 F N 10 / 56 1G H 0 1 / 1 2 "!*
s ≡ sab = (Pa + Pb)2= (P1 + P2 + P3)
2,
s1 ≡ s12 = (P1 + P2)2= (Pa + Pb − P3)
2,
s2 ≡ s23 = (P2 + P3)2= (Pa + Pb − P1)
2, F"!",G
t1 ≡ ta1 = (Pa − P1)2 = (P2 + P3 − Pb)
2 ,
t2 ≡ tb3 = (Pb − P3)2= (P1 + P2 − Pa)
2.
1 7 0 C 1 6 1 0 C O 7 C F"!",G 8 / 1 C 110 F"!",GO 7 ta20 tb20 ta30 tb10 s13 F QARG
0 1 1 FPi · PjG0 i, j =a, b, 1, 2, 3 C C F"!",G
80 1 / 5 2 K0 1 1 2→ n 11 F2→ (n−1)0 F1→ 2GG F 1 1 2 3;$1 Q RG 1 2 → 3 7 2→ 2 1→ 2
1 1 F"!*G?a+ b→ 1 +X 0 X → 2 + 30 2 X 7 5 √s2 7 "2 11
1 t1 F 12 1 C X /1G 7 7 s2
1 (t1 , s2) 0 2 90 F 2 G 2 . 1 / 7 1 0 156 1 2 0 5 ! /01
"-E
H "!"! % O!U!+ !0 t1 = (ma−m1)
2 0 . JVL
# +%9 B%:,3%. 1&,2,425
2 2 L4 C 1 C0 7 1 2
C / 2 2 K?
R3(s) =
∫d3p12E1
d3p22E2
d3p32E3
δ4 (pa + pb − p1 − p2 − p3) . F"!"*G
;1 C ?
1 =
∫ds2
∫d3p232E23
δ4 (p23 − p2 − p3) , F"! !G
2 E223 = p2
23 + s2 7 C F"!"*G0 1?
R3 =
∫ds2
∫d3p12E1
d3p232E23
δ4 (pa + pb − p1 − p23)×
×∫
d3p22E2
d3p32E3
δ4 (p23 − p2 − p3), F"! "G
R3(s) =
∫ds2R2
(s;m2
1, s2)R2
(s2;m
22, m
23
). F"! G
"-+
C 0 F"! G 2 1
R3 =1
8√sP ∗
a
∫ 2π
0
dϕ
∫dt1 ds2
λ1/2(s2,m
22,m
23
)8s2
∫dΩc.m.
3 . F"! AG
H "!"" % O!U!+ !0 t1 = (ma−m1)
2 0 .+ mb = m2 = m1 = 1 1 0 mb4+ m5 = 5 m3 = 2 s = 60 1 JVL4
=5 ma + mb → m1 +√s2 C
?
• 19 C s2 t1 O
• 1 1
m2 +m3 ≤ √s2 ≤√s−m1 ; | cos θ∗1 |≤ 1 . F"! BG
> 15 2 2 F"! AG 81 1 F/ C QARG0 19 2 1 2→ 3?
s±2 = s+m21 −
− 1
2m2a
[(s+m2
a −m2b
) (m2
a +m21 − t1
)]∓∓ 1
2m2a
[λ1/2
(s,m2
a,m2b
)λ1/2
(t1,m
2a,m
21
)]. F"! -G
"-,
C 0 2 s20 C56 211 1 0 C 11 2 L40 t1 θ∗a1
C 2 L4 0 1 t10 s2
d3R3
ds2 dt1=
π2
4s2· λ
1/2(s2, m
22,m
23
)λ1/2 (s,m2
a,m2b)
. F"! EG
8 0 2 L4 1 (t1 , s2) 6 0 15 ?
λ(s, s2,m21) ≥ 0 , λ(s2,m
22,m
23) ≥ 0 , F"! +G
# <%:,3%. 1&,2,425 ,2,C 1,2,,3
F"! EG F"!"AG 1 2 0 C 1 / C ? 1 C F"! EG 1 F"!"AG0 2 s s2 F a+b→ 1+2+3 6 5 2+3→ 1+a+bO1 6 s1 1/ t1G C F"! EG 2 9 / Fg/G1 / a + b 1 / 2 + 30 0 1 / 0 9 2 + 3 ./C 9 C L &&& 1 C 2 ”1” 2 1 F FB*GG 8 0 7 / C
0 2 BA 56 7 5 C 0 56 / 90 112 C 9 7 2 2 7 /9 FB+GO 112 (1 − α)−1 1 F 0 1 2 1 G 8 0
"-*
95 FC 9G . 0 156 7 ” 1/ 11”0C 11 / 2 / 1/0 9 22 5 / / F/G
8 0 9 95 Fn+wq_xbcG0 0 60 FnT G0 1 5 1 71 F2 1 C 1 2 1 2 1 9 1 G0 C 1 FnT G 1 F*""G F*EG01 6
σ =
∑
∫PhaseV olume
| < f |T |nT > |2InvF lux(nT )
. F"! ,G
7 0 C 9 0 7 C 1 / 71 / /. 7 1 InvF lux(nT ) C F" " " < 6G
1 6 20 C / 1/ F.0 1 1 / 2 0 0 1 2 10 2 C /6 1/ 0 112 G.7 0 0%'0 FnT G 9 1 5 1 5 FnT G ? 2 5 122 F1 G 0 %'0 7 9 C0 2 2 10 1 10 2 C 20 C 1 F"! ,G0 1 F" " < 6G?
∑
∫PhaseV olume
| < f |T |nT > |2 =
= σinel(nT )× InvF lux(ntransfT ) . F"! *G
"E!
I1 0 156 2 2 2 BA 1 0 1 FB,G 1 F"! ,G0 1 F"<6G0 InvF lux(dT ) 1 1 7 2 1 9
R (n, d) =InvF lux(ntransfT )
InvF lux(dT )=λ1/2
(sn,M
2targ,m
2n
)λ1/2
(sd,M2
targ,m2d
) . F"!A!G
; 12 1 15 2 2 6 2 "!* ” 1” "!" 2 1/C 2 C F"! EG? 2 1 a b 10 1/ 5 ”1C ” 0 1656 2 3 7 1 C F"! EG
H "!" % ” ! ”/ 0 ! 0 !00 O 0 a b”.7” !0 Xab
√s
0! 1 0! X2 √s2
0 "! 0 0 2 3 #! !00 ! !00 .7 R2 . ! .7 00 a + b → 1 + 2 + 3 + ... . 0 !" !00 .7
"E"
)& )
&
. "
" *
/ 1 C/ 2 K0 2 L4 7 8C / /0 56/ 1 1 1 .9 7 0 1 2 10 / 6 / 2 /
H 0 15 7 2 7 1 2 1 1 2 2 M 1? M → π+π−π0 7 1 < 3π | T | M >= Mα (pi , Ei)0 2 α 1 % 1562 1 "0 1 5 F 2 1 7 C
"EA
G0 F"0 !0 "G N 1 " 507 ! ! ( 2/ / (E1 , E2 , E3) 1 2 0 1 72 1 1 F "!"AG
. /560 1 2 0 1 1 1 0 7 C 5
Mα (−pi , Ei) = (−1)PMMα (pi , Ei) , F"""G
2 PM N 1562 ; 0 1 2 5? Iin = 0 ;
1 2 C C 5 C1 F1 C 2/ 0 1 1 C0 22 G0 /1 1 1 C 1 1 ?
| 3π , 0 > =1√6
[| π+ , π0 , π− > + | π0 , π− , π+ > +
+ | π− , π+ , π0 > − | π+ , π− , π0 > −− | π0 , π+ , π− > − | π− , π0 , π+ >
]. F"" G
0 /1 11 &%' ' > 0 F"""G F"" G0 1 C 2 7 0 0 1 2
0 1 7 < 3π | T | M >C 2 / 0 0 1562 M F0 7 01 N 7 1 0 N 7 G 0 / 56 /1 O 7 72 U / 10 0 1 CMα (pi , Ei) 55 M 1
8 $ 9: 9:
"EB
%41%/ :,% 4, 41, # % 2+ 1,%
7 7 M (pi , Ei) C 72 2 0 1 0 1 1 1 /5 1 10 56 F"" G #C 17 0 7 C C 72 < C0 7 C 1 E1 = E20 E3 = E20 E1 = E30 0 2 1 C 0 56/ 7
6 0 56 1 0
M (pi , Ei) = (E1 − E2) (E2 − E3) (E3 − E1) · f (E1 , E2 , E3) , F""AG
2 f (E1 , E2 , E3) 1 1 5// 2 . 5/ / 2 C F 0 / 0 0 5 7 / C 2 G .1C0 7 1 C 1 2 0 56 2 72 2 0 C 1 1 I2 27 / 1 1 1C F""AG
1 1 72 1 r ϕF 12 "!"AG0 1 2 C ?
dρc.m.
dS= :`lkc · (E1 − E2)
2 (E2 − E3)2 (E3 − E1)
2 =
= :`lkc · r6 sin2 ϕ . F""BG
8 5 0 1 2 2 0 1 5 1 / / nπ/60 n = 1, 3, ..11 FE G / nπ/60n = 0, 2, ..10 1 5 F """G
@ 0 F""AG 0 7 1 C/0 92 M C F"""G
"E-
C 11C 0 C C 1 A 1 7 0 1 1 2 0 1 2 C F 5 1 11C5G0 7 1 1 / 1 I 6 O 10 7 1 / / 1 / 1 5 7 C 6 T > 0 0 ' ) *& +1+ 6 *
%41%/ 32,+,8, :,% % 2+ 1,%
7 1 2 "0 2 −1
F"""G 2 / M0 7 C 1 I 1 0 1 1 / 1 10 C 1 O
M = [p1 × p2 + p2 × p3 + p3 × p1] · f , p1 + p2 + p3 = 0 , F""-G
1 O f N 1 1 / 2 9 F""-G 1 11 1
M = 3f · [p1 × p2] . F""EG
. 2 1 7 2 2 7 0 1 0 f1 9 2 0 C 1 0
dρc.m.
dS= :`lkc
λ(p21 ,p
22 ,p
23
)4
, F""+G
1/ / C5
dρc.m.
dS= :`lkc
[1−
(1 +
2ε
(1− ε)2)r2 − 2ε
(2− ε)2 r3 cos(3ϕ)
], F"",G
"EE
2 ε 1 F"!"+GC 56? F"G / /
0 1 2 2 1 O 2 5 7 F 16 G I 0 2 2 1 6 2 F G 1 2 2 FAG 2 2 1 50 N C F 6 F"",GG 0 1 6 2 2 """ 7 2
%41%/ 143/,32,+,8, :,% % 2+ 1D
,%
7 1 "0 2 +1
F"""G 1 2 / M0 7 C HC1 C0 16/ /0 C 1 0 7 0 1 1 1 0 C
M = f · [E1 (p2 − p3) + E2 (p3 − p1) + E3 (p1 − p2)] . F""*G
. / 72 10 7 C C 1
M = f · [p1 (M − 3E2)− p2 (M − 3E1)] , F"""!G
0 2 1 6 F 7 E1 = E2 = E3 =M/3G $ 20 F""*G 0 2 2 /0 2 p1 = p20 p2 = p30 p1 = p3 1 C 6 F 7 G @ / /0 2 p1 = −p20 p2 = −p30 p1 = −p301 F "!- """G
"E+
1 1 0 / 16//0 C 1 ?
dρc.m.
dS= :`lkc
[1− 2
2− ε r cos (3ϕ)
]r2 . F""""G
H """ % % 0− 1+ 1−
JYL
"E,
%& " *
= 0 16 % 0 2 /12 12 0 2 2 12 1 2 0 0 1 f 10 ”1 ” 0 2 /1
H "" # % 0 J(L
$5 / 2 /1/ 1
"E*
C 0 10 2 Q R F "" G @16 1 C O 0 2 15 0 5 1
. / 15 1 / F / / 2 1 MG0 1 C / 1 I0 1 J P 0 (I JP )0 0 / 0 2 16/ 2
. C 2 L h&0 C9/ ? 1 1 F1 G0 2 7 1C 92 F G 2 K I 1 15 0 1/0 7 C 0 1 9 0 /0 / / / 15 1 6 / 2 7 / 1/ $ / 1 C /
"+!
3+
""
6 15 71 / 7 1 2 12 0 //0 2 1 2 71 F1 0 1 0 /0 1 1 / G 2 C
$ 1 2 K 2 0 1 / 1 0 12 2 / F1 9G 1 1 / 2 0 5 71 . 7 C 1 / F7210 / 1G @ 1C / ”” 71
1 1 1 1 FL *G 1 1 /
+ +$ ;)<
"+"
1 1 1 9 2 1 F KG 1 9 F10 2 5 G0 C 1 1 2 0 2 / 0 1 2 K 2/ / 1/ 1 80 ”1 ”0 ”” 1 Q R0 1 1 / 0 1 C $ 7 / ”” 7 0 0 D ; 0 1 29 7 0 F 9 G 9 21 D 7 1 2 1 0 17 1 1 1 56/ 1 10 1256 F 256G /C 2
81 1 0 1/ 71 0 56/ 7 / 0 ” 5 1”0 2 55 71 ./ 7 1 0 56 2 1F0 10 /9 ”” Q+"RG> 1C 5 2 C 1 2 / 1 1 7 / 1 8C 1 o Q+"R
!"$
1562 9 2 0 1/ 1 0 0 C 2 1/ (0, 1) /
3 9 / 2 56?
" 3 15 /9 1 2 0 1 1 0 2 C
"+
56 1 162 /9 1
> 0 1 / 1 / 1/ 20 1 1/ C 51 / 0 C 25 2
C 1 15 0 15 2 1 / / / 2 0 1 2 5 0 5 C 2/
A 2 2 2 / 0 1/ (0, 1)0 1 12 11 1
<0 ”1 ”0 / 92 .7 5 % 8 5 C0 2 1 5 1 2 1 97 2 10 1 2/ 0 6 *7 !0 8 *&; 1 0 1 2C 1 0 / 29 1 2 I 2 C 0
=& "$ )
$
.0 9 56/ 1 1 2 0 7 1 C Q+"R. 9 10 0
8 1/ 56/ 1 71 $ 1 2 x0 1
"+A
[A,B] 1 F10 71 G f(x) F f(x) N 1 G
80 2 2 0 C 6 2 0 2 1 7 9 1 7 56
H [A,B] N 1 O 1 i xiO x0 = A ; xN = B 20 x1C 1 i ∆Wi f(xi) · (xi − xi−1G0 ?
∆Wi =
xi∫xi−1
f(x)dx ;
N∑i=1
∆Wi = 1 . F" "G
: 6 ) * + ∆Wi 0 [0,1] F1 0 1/6 " 1 15G 80
1 w0 2 56 12 F W[)yG 11 j F 5∆Wj−1 < w ≤ ∆WjG0 56 x xj F (xj+xj−1)/20 C 2/ 11 G > 0 5 2 12
20 1 10 0 11 5 1 x0 22 12W[)y 0 w(x)0 7 N 1 N 0 0 92 25 FG 1 $ 20 2/ 0 15 1 1 / / 1 /
; 0 N 2 1 y 156 /C 2 x1 1
y =
x∫A
f(x′)dx′ ;
B∫A
f(x′)dx′ = 1 . F" G
! $ "( ( (0, 1)
"+B
C 1 1 12? 5 5 15ξ = g(x)0 g′(x) = f(x) 7 dξ = g′(x)dx = f(x)dx02 ξ 1 [0, 1] . ξ 165 2 W[)y0 / 9 g(x) = ξ 20 1 1 x0 1 2 1 > !
$ 3 * ; .5 . 1 1 exp (−x/X0)0 2 X0 1 2 0 56 0 11 7 1 0 N0 ”9/”2 C 9 2 Xmax
;1 2 1 ;?
dξ = − 1
X0e−x/X0dx ; ξ = − 1
X0
x∫0
e−x′/X0dx′ = 1− e−x/X0
x = −X0 · ln(1− ξ) . F" AG
I1 2 2 ?
! ! " #"$ %# " %% & '
( ! ) * +,)'-./0*1 +,) " 2 '3.450*&+,)16 0&57&1 '*8
9 ! ) * :"
9;(6< =>;?>@AAB
$ * ; ;2 10 9 2 / ”” C 0 6 0 . 2
8 + = &( / ( ( = (
"+-
* ! 2 1/ 5 7 2 1
; 0 1 1 0 1 1 X = 0 1 1 σ = 1 F7 1 / 25 6 G0 x 12 1
1√2πexp (−x2/2) . F" BG
1 F” 1 ”G 1 (x, y)0 2y N C 0 1 1 C 0 x0 2 I2 x y (x, x+ dx; y, y + dy)
dWxy =1
2πexp
(−(x2 + y2)/2)dxdy =
=1
2πexp (−ρ2/2)ρdρdϕ = dWρϕ . F" -G
C 1 2 1 1 ϕ 1 56 1 ρ?
dWρ = exp (−ρ2/2)ρdρ⇒Wρ = exp (−ρ2/2) ; F" EG
$ 0 1 ” 2 1”?Wρ C (0, 1) H2 Wρ
1 7 0 / x y?
ρ =√−2 ln(Wρ)
x = ρ · cos(ϕ)y = ρ · sin(ϕ)
F" +G
2 ϕ C 1 (0, 2π) 20 1 1/ / x y0 20 7 T
$ / " $
"+E
4'
%& "
( 5
'> ” $”
H K 71 C 5 / 1 0 165 / 2 % 20 1 5 FG 71 C 52 K 8 /1 7 N 2 / F 0 0 N G? 1 1 2 71 ” C ”0 C .7 C 2 1 20
/ 6/ /0 1 2 C/ 71 5 0 56 6 7 1 C62 0 > 7 1 C FG 0 1C I0 1/ 0 C
"+,
2 1 2 I 0 5 C
7 ?
" 5 0 6 O
1 5 0 1565 1 2 6 2 56 7 / / 0 1 N 1//0 / 7/O 2/ NC 1 F10 2 2/ G / 0 5 5 F 0 1 0 G
A H2 56 7 0 156 12 6 1 6 22 0 156 C5 > 1 C 5 F1 10 N C 0 1 N N 10 7 1 1G0 7 2 F 7 2 G0 2 F 7 G
B 1 0 2 / 1 1C 2 562 7 / 20 2 2 1 C 1 F9 1 0 7G 1 F1? 1N 1/ 0 1 2 11 0 1 7 0 15 1 G
. 19 5 1 > 7 1 / 71 / 0 9 1 15 /0 62 122 / 1
"+*
$ F210 11 0 11 0 2 G 15 65 0! 19 F 10 CG 1 0 1 0 C 1 C/ 1 7 9 C . 6 0 C 0 2 ;2 1 ” ” U ” ” ” ” 126 F G72O 1 7 / 1 1 C 2 5 C5
0 ” ” 72 F1 N 1 ? 1 192 7 2 2 1 19 G0 1 1 F GO ” ” 1 72 7 O 1 71 0 1/ /0
; 0 2 0 C 10 2 0 ” ” 1 5 1 F4 1 1 2 2 1 2 0 5 C 0 1 2 2 7 2 N 1O 7 5 C 0 G
10 2 1 FτG L F 1 C / 2 / 0 C G0 0 1 1 C m 2 ?
m = p ·√( c
L· τ)2− 1 , F"A"G
2 c C 15 11 1
1 / 1 1 8 2 1 C
5 $ 50 1 20 9 F 7 2 C
",!
H "A" # ! . ! 0 JS(L
H "A T ! 1KVK4 @ ' ! ! 0 1/β JS(L
1 7 LG0 15 C 2 2 21 0 2
;1 1 1 0 0 0 71 5 5 7 10 1? / 156 1 0 1 1 72 5 5 .7 0 11 / 72/0 1 ” 2” 2 5 0 10 5 0 1 80 7 11 5 1 .1C / 72/0 2 0 / 1 6 2 0 5 F1G 1
","
H "AA # ! 0 ! 0 1 KVK4 ! . JSVL $ ! 0
0 / F7 2 0 20 1G 15 1 1 2 / 1 / 1 8 C 15 1
7 1 N 2/ / 0 / 1 /2 ; 0 /2 1 1 71 0 1 C 6 71 / / / 0 6 6 5 6 2 1 . 71 N 1 K 1 / / / ? 0 2 0 /0 0 / 1 0
71 15 1 0 1/C 1 C H F/ 1 11 / 1 / G 8 C 0
",
1 / / 0 1C6/ 50 6 1 % / / 7 C
I /2 71 /9 1 75 2/ 32 / ”1/ ” FIH0 ”cdjb ^_`ab:cd`l :zqjb_k”G
H "AB .1 / 1
$ % DEF 1 "AB . 0 1 0 11 1 C 2 1 7
8 7 K 0 7 565 K 2 2 F "ABG > 2 0 9 199 C 0 5 1 50 1 1C 5 7 2 1 K 1562 7 K 2 > 0 1 1 15 F 7 2 1G
. 7 115 2 $ 10 7 11 H C 1/ 9 0 2 / 1 1 2
1 7 56 2 0 11 0 1 > 10 1
",A
0 1 F 11 G .7 7 1 5 11 . 7 1C 16 2 20 2 I 0 1 16 2 0 15 (x, y) 0 12 1 / 1 1 2 11 / 7 7 0 0 / 1 7 1 2 16 1 15 ” C ” 562 1 1 1 0 1 5 0 9 0 1 5 / / 1 F0 7 C 0 1 5G
"A- 1 1 71 1 7 2 / F'|_dlx,0 <1G # f|' 7 1 1 ; 2 FWY)|G 8
. 2 F "A-G 1 1 1 7 1 .56 K 1C C 90 55 1 w|Y H / 7 1 (x, y) 1 2 56 0 F 1 G C 1 20 2 2 F 9 7 0 9 1 2 G
C 15 15 110 1/C ; 0 2 1 2 C ? 2 10 1 2 7 7 2 8 C 68 09 w|Y N 1 F 1 1 222 20 2 1 20 C G0 F
",B
H "A- . 0 2 2 1 f|'0 72 1 0 γp
1 1/ 0 2 CG0 2 w|Y 1 1 10 1 1 C 1 0 2C 1 0 w|Y5 5 5 6 17 7 1 C C 2 0 0 2 1/ 9 ” ” 7 2
2 0 29 1 9 w|Y 1 95 1/ 150 7 7? 2 / 5
1 2 Q+BR
",-
-#> )
& &"
1 0 019 1/ > 9 1 F 15 N /2/0 1 71 G0 0 9 9/ / 7 / / 1 10 1 1 6 ” /”
/ C/ 0 1 5 95 0 9 0 56 F7 6 0 G 1 / / 1 /0 ”” 1 / ; 0 56 F$JG 2 1 C/ / / 720 C 6 1/ 12 1 / 72/
30 1 71 2 56 $J N .7 C / ? F"G FCG 0 F G 75 F10 22 1 2 C /0 2 C 2 1 G FAG 1 2 / 71 FG 5 5 5 F "AEG
86 7 / 5 0 C / 1/ C 5 11 2 / / 1/ 1/ 1 7 / / 10 5 12 C 1/ C? / ” ” / 1/ 0 12C 0 2 / / 1/ /012C/ 565 $J F20 0
< ( / ( $ 0" &( ( ""
",E
H "AE .1 1 56 $J 1 / FC 1 1 G 1 2 @ F1G1 0 2 0 56 56/ 11 0 C 12 O 1 / 1 92 C 56/ F 2 F G 1 / G
1 9 2 G .7 6 ”2” F2 / 20 1 C 0 1 U 9 GO F 2 G 75 $J 6 C 1/ C0 1
; 56 F$JG 1C 2 1 2 F
",+
/ 2G /C 20 56/ F 20 2 1/ 1/ N 6 1/ G
2 1 C 0 6 5 56 562 $J 6 ? 6 0 1 6 0 25 1 1 C0 2
. / 72/ 1 F G 1 2 /C 9/ 1 U / 1 ; 7 C 161 165 C/ / 72
8 256 2 1 0 6 / 2 56 F$JG
20 0 1 0 75 2 1 C ” ” 0 26 21 56 2 / 9 / O 1 5 29 21
0 1 56 $J 0 2 5 2 2 7 2 F 2 56/ 2 / 0 / G . 0 1C 2 2 10 10 0 2 2 6/ F 1 1 90 1/ 2 G? 11017 2 > C 1 9 C5 7 0 1 5 2 "AE C 0 1 0 1 "!
; 0 71 / / 2 1 0 56 12 9 C9 2 2 1 C/ / 72 .7 C C 0
",,
20 C9 0 / 72 1 561
• 1 / 1 /0 . / 1 0 N 25/
• . / 1/ C 2 56 . C/ 2 U 2 0 5 $J 9 0 5 0 72 F 0 G
√sNN = 4 − 9 37 0 10
Au− Au U − U 0 C / F1 G 56 /
• 7 /
• 8 2 6 C0 1 1 / /C
• ”> ” 0 1 C F 1 G 21 QE"R
• H 1 / 2 1
• 10 / / F G 2 56/ 1/?
= ; / / C/ 7 /
= H1 6 / F1 0 GO C O 1 0 1
= / C / = 1 C 1
",*
• < 1 ”/” ”/” / C / C F0 G0 F 9 G
• 1 1 7 / O 7
• 2 2 F 15 N /G
• / C / 12 2 1 2/ 0 // 0 / 72
$ % ' 3!';
C F1G / C9/ / / 0 / C0 0 0 0 10 1 7 10 0 10C/ / F 1 11. / = %>9 G 50 0 1 7 0 5 956 C 6 C
2 1 2 9 0 1 ? 1 K 0 1 1 K 1 > 192 80 1 / 72/ 1 7 5 0 / 1 1 12 0 0 1 1 1/C 1
H9565 0 71 0 5 1/ / 72 2 12 /2/ 12 1/ 1 1/ 9 ;?
• %2 / C / 1/ C 1 1 1
"*!
• .2 1 71 C 2 1 F C 0 7 / /C 1G 1 /720 1 156 2 5 1 / 1 F1? ” ”G % 1 9 1 019 6 1 1/ 71
• ; 1 ” 1” 9/ /C / 1/ 9 1 1 9 9/ 15 1 2 6
20 ” ” 1/ 1 / / 72/ 6 1 5 9 / / 2 192 I 5 0 560 2 1 /72
?@
1 C 0 56/1 / F 1//0 1C /G 72 71 / / 1 F 0 G0 1 72/ 12 C
C9 2 2 1 C/ / 72 71 / / 20 1 0 56 0 1 2 / 1//0 9 0 F /G
; 1C / 72 5 0 C 6 C 2/ 9/ @ C5 ?
"*"
• H 12 / C / // 72? 2/ F5 C 0 10 1 0 G
• . 1 9 1 7 F1 1 / /G? 1 ? 1 F/61 G
• . / 1 / 1 2 0 N 1
.1 7 / 15 ? F5 7 5 7 2 0 15 5 12 / G
?* 6 9 G
# 0 / 1 1C 0 C 9 6 0 / 1 2 C60 7 0 7 1
80 C 1 20 1 0 1 ” ” D 9 F G 0 C D $ C 0 11D
. ” 1 ”0 C1 / 1 19 9 / 1 0 C / / 1 09 0 7 2 9 2 2 1 / 1 95
"*
1 ?” C
0 1 5 / /20 C 9 7
I C 1 / 9 2 6 1C 0 1 1 ”8 1C ” 9 2 / /
0 C / /2 $0 / 6 1 / A % * * ' 7 ' %* 0 A4<:7 ! %' 0 7 * ; ”
6 F 1 = A- C 1 / 72 F H G0 9 .C 5 !"! 2G?
”-7 !% 0*% 7 & 8;”
> 1 5 2
1>:>>1 )#?@#?88 & 8! $ $/$ 8A? B C$ (" . * $
"*A
%"6
7+ !
45 71 C 56 56 7 / 7 N 15 N 06 56 71 $ 10 C 9 C / 1 0 5 9 / 0 2 2 5 9 2/ 0 /0 C 1 / 1 1 9 C 112.7 C 7 1 1 1 0 ”1 ”@ 0 0 C /0 1 5 > C0 1 1 1 6 9 0 2 1 9C C 2 0 2 1 1 2/ 1 0 2/ / 5 10 2 12 1 2 1 0
"*-
9 $ 2 / /0 ”1 2 95 ”? 2 10 / / C C .7 7 0 C ? 50 7 2 2 1 2 1 > 1
$ C 20 C 2 / 720 72 1 0 156/ 165 90 9 "! ! 37 F G 8 71 7 72 5 7 1
# / 72 15 9/ 1/0 7 1FG0 5 5 / 1? F2 5 / 1 C 72G F2 72 56/ 1 0 1 6 5 G /0 7 10 1 16 9 F ” 71 1C 95”G 1 1 1 0 2 / F 1 C 1 1 G / /F / 1 N 0 1C 156 C C 0 1/G $ 10 / 72 N C 1 2 / 720 1 C 1 ”%9 ” FfXYG > 1 1 ”C ” F&fYG
8 0 5 5 0 0 71 2 1 .7 5 2 / 2 $2 C 15 0 5 C
3
2 ”0"” " $"$ DEDFG
"*E
0 71 F5 G / 0 1C 56/ F ” ” 10 2 1 0 0 72G . 2 0 2 71 / 10 71 / / / 15
. 72 2 56/ 1 E∗ 1 2 9 56/1 I2 1 s F 1 72 G s ≈ 4E∗2 20 1 C s / C 0 02 / F””G 0 2 F”9”G 1 F 71 1 5 71 ”1C” 95G0 C 725 Ebeam 2 1
Ebeam ≈ s
2mtarget= 2
E∗2
mtarget, F"B"G
9 F BG I 0 C2 2 C s 1 / ”” 95C 2E∗/mtarget 6 0 / 1 / 2 0 / 71 C / F G 2 0 56 1 0 1 0 ? 1 / ”” 95 2 1 6 90 2 C / 1 2 1 9 / 11/ 1 1I/ 9 7 C ? 2 1 71 2 1 1 /71 0 20 0 71 / 1C9 ; 17 2 71 2 7 / 1
, $" ”HIGJ KLMNGK GIOGMPQGRKS”
"*+
$ ;
8 C9/ / 52 N 10 1 1 F G / 1 11 2 / ” 22” 10 N * 8 0 1 ” 95” 1 1 5 ” 1”F”m\co pq:c`_”G0 5 1 6 9 F F,+GG?
L = Nnucl · Iτ· δ = l · ρ
A· Iτ· δ · 6.022 · 1023 × nmol =
n
σ, F"B G
2 NnuclN 9 2 1 10 IN 1 0 δN ” 1” F”m\copq:c`_”G? δ = τ/T F τN 1 0 TN1 1 7 / 1 F N /GGO lN 69 F G0 ρN 1 6 9 F 2U3G0 AN 6 9 F 2UG0 nmol N 0C6/ 9 6 90 nN 2 1 FG0 σN 2 11
8 / / 2 F"B G / 90 2 126 F1/ U2 /G C 1 8 1 11 126 0 C 20 2 0 F"B G 1 1C / / 0 10 1 1 71 6 11C0 F"B G0 5 0 9 1 5 1/ 10 11 9 112 1
= > ?@12· 5O 17 0 1 C 15 " 3 F G I "B" / 1 / / 101562 9 6 " 1 δ = 1
; 1 1 µA mAO 11 0 1 µA
"*,
I "B" 9 6 " 1 δ = 1
6 ρ0 2U2 [ fF&~"!6UG σF" 3G
H2 !!+!, B ·1028 B
FCG
CH2 !* !*- "B B! ·1028 -
F2G
[e + + E! ·1028 "+
| ""A- !+ AA·1028 AA
≈ 6.25·1012 ≈ 2π·1012 U ? 1012 U ≈ 0.16 µA ≈ 1
2π µA
2 "&
%9 56/ / 72 / H / FG 1 2 Q -R0 ?
F"G F7 U1 G1 XW[ F30 320 'G 72 10 0 +- 37U × * ! 37UO 1 7 U1 C 1 5
F G 1 1 wh[wW) F=#40 V 72 1 " I7U × " I7UG
FAG 1 1 fXY FMH0 VO + I7U × + I7U 5 !!* 2
"**
56 6 2/ ?
• 7 1 ||&& 1 * 37U Fb−G × A" 37U Fb+G 5"!33 :−2:−10 7 F'f[Y0 VGO
• 7 1 $$ <1 1 , 37U Fb−G × A- 37U Fb+G 5 "!34 :−2:−1O
• ; / 72 F.0 $ G0 1 7 1 721 , × , 37 5 "!31 :−2:−1O
• WX&Y %/ FVG0 1 C 1 1 F -! 37U × -!37UG 0 1/ F1 1 1 G
; / 0 /6/ H0 C C 1 ?
• '|'0 C 1C 1 fXY0 6 1C 5 2 1 1 B!! 37U F C1 1GO / 2 1 MH1 "B"O
• 9 5 1 |[WY <1 1 -! 37UO
• 56 "*E! 2 [' FAA 37U0 %/0 VG
!!
H "B" # 1 MH 86 /
!"
H / 72 1 6 5 ?
• 1 / ;=> F. 0 72 1 +! 37U F "B G 8 16 "*E+ 2 % 15 5 2 C ;=>
• > 1 >..B F;<= % O E 37U × E 37U $ >..B16 "B "*+* 20 "** 2 >..B FH "BAG
H "B 8 1 #+! ;=>0 .
!
H "BA >..B? 6
!A
A $ "&
"
7 / 5 1C / 720 56 HO / 7 2 2/ / 8 56 90 5 71 1 1 / 1 # 7 1 5
. 0 1 ”1” 1 F7 1 G0 C 1C / 72 > 1 3? / 9 7
L 9 1 1/ 71 0 9O C 7 1 0 F 1 0 1 0 1 G L 0 C 9 / /0 1 1? 2 / / F 1 G0 71 1 F5 G / 7 D
64230'E 3 ,44 04,+2&
8 1C / 72 H 8K ; </ ; F8;<;G0 "*-E 2
> *% 1?2 <4H4;
!!A 2 1 4 / 72 F4=> %G 8K2 ; </ ; / ? / F 9 / ”2” 0 1 1 -! G N 12 1 562 /162 / C 0 1 2 C
!B
2 C 0 1 21 9 1 0 ”2”F Q**R I"B G
I "B I/ / 4=>
. / 72Tmax 1 0 37 9 12 72 Z/A = 1/2O Tmax0 37U[ 4 6L 1 F"UwGU !" !" ÷ "! F” C”G τ 0 !- "!= 1 FδG !!- ∼"0 10−6 ÷ 10−7 10−10 ÷ 10−11
. 6 0 , "- 1 1/0 I "" !. 0 !+" -"-$ /110 9 *E$ /110 9 EB1 10 ""!86 /C 0 ∼,!6 2/CC 1 B-0 ×"E
$1 4=> 2 / 10 5 1 1 0 1 8 5 7 2 2 1 1 1 90 C 1 71 1/0 1/ 1/ I 9/ / 1 "BE 1 16 / 1 2 10 I"BA 1 1 / ; 1 6 5 &O && 1 0 17
!-
I "BA .C 1 1 2 1 4=> F UGO ∗ / 1 0 1 / / 1 1 d→ p 0 1 FPG N 0 C 1 71 5 2 1/
. / Z F&G F&&G
p 4 · 1012 1011 1013
n 1010 5 · 108 1011
d 1012 5 · 1010 1013
dpol (1− 5) · 109 2 · 1010 FPG 2 · 1010 FPG∗ppol ∼ 2 · 106 ∼ 108 ∼ 108∗npol ∼ 106 ∼ 108 ∼ 1083He 2 · 10104He 5 · 1010 5 · 109 2 · 10127Li 2 · 109 2 · 1010 5 · 101212C 109 7 · 109 2 · 101216O 5 · 10720Ne 104 2 · 108 5 · 10924Mg 5 · 106 3 · 108 5 · 101128Si 3 · 10440Ar 3 · 107 2 · 10956Fe 101165Zn 5 · 101084Kr 2 · 107 5 · 101096Mo 1010119Sn 2 · 108131Xe 107 2 · 108181Ta 108238U 3 · 106 108
C F G !! 2 19 6 1/ 2 1 1/ 1 71
6 1 8;<;? C/ F1 ;$G $ 7 2 1 B
!E
0 2 4 6 8 10 12 14 16 180
500
1000
1500
2000
2500
3000
3500
4000
положения деполяризующих резонансов(Нуклотрон, дейтроны)
Ξ+2K
Пороги, достижимые на ускорителях протонов(промежуточные энергии)
φ
Ω+3K
DC12C d p
COSY
Nuclotron
рожд
енны
й в
реак
ции
избы
ток
масс
ы, М
эВ/c
2
кинетическая энергия пучка протонов (в л.с.), Tkin, ГэВ
H "BB %! 0 pp p d p 12C -./ . 0 3 !01 1 4 3 ! 0 3 !4 . ! .!" @! d !01 .! JK))L4 %! 3 ! O CC
H "B- M0 *! 0 pp %! 3 ! O CC
!+
H "BE 86 / 4=> 8;<;
H "B+ > F1 2 11G C/ 1 2 /
!,
-' $4H? )+;
;5 .;<= . 2 ; < =0 2 1 F 72 1 " 37G C .;<= F Q"!"R 1 C GO C 1 .;<= . / 1 I"BB I"BB"BE
I "BB C9 1 / .;<=
M 60
% PY,
6 ! ,)
2 ! KSY T
-! !" Y &
O *)P) N
O !" 3 V)KV 2
2 !" ,)) &
8 ! !0 VµA
V)e
”Z ” ,)e
&!! K)−6
9." Y)))
I "B- . 1 / 1 .;<=M!0 Tkin N3& ∆T/T e 8KW
@0
K K < 6 · 10129 p!0
K K 108
2 !0
0 V, 2
K 10−1 108
0 !0
K K 1010
& p!0
! !
!*
I "BE . / 1 .;<=
O 8! ∆p/p e 8 M0
23&W KW
-0
π+ *,) P 106
π− *,) P 3 · 105-0
π+ (,) (,'K( 3 · 106 − 1.6 · 107
π− (,) (,'K( 3 · 105 − 106
µ+ (5 K( 3 · 104?
µ+ KS, K) 3 · 105 !0
µ− KP) K) 9 · 104
I "B+ 1 .;<=Tkin 8 % O
KW N
10−2 3& ' 100 23& 3 · 1014 K) ,)
1 /0 1/ 7 0 1 1 9 12 1/ / 12 12 C / 0 11 2 6 / / @ 6 5 2 72 21 F 16 "*E+ 2 9 71 C 1 "*+! 2G? 1 72 "37 / 1 12 12 F C / G /9 1 3
> * 4B2? )@ +;
> / C 8 1 0 ;I>=0 1 1
"!
+!372 ;=> F. G 1 1 72 "! 370 1 C 10 1 4=> 8;<; . 1 1 1 0 2 1 / FG = 12 7 5 9 71 1 1C / 72 / 1 1122 C 1 7 1 1 2 / C/
. 9 1 1”/” 1C 720 56/ H
64230'E 3 ,44 04,+2&
> * FIJK )L'7 +;
# Y' S/ N 9 7 /C 1 2 1 72 1 F G - 37 6 ,A 370 2 9 12 C φ 1 1 / / # 1 C 1 1/ 1 81 1 I"B,O 1 7 2 C 0 10 Q"! R 1 C >1 0 0 90 N 9 1 1/ 1 ; / 1 F10 1 5 8;<; 1/0 C 0 I#H&&0 "B AG0 C "B,
""
I "B, 8 1 Y'
. @
pmax 3.30 37U
p actualmax 3.65 37U
δp/p 0.5 · 10−4
;
2 1 2 · 1011;
2 1
F G 5 · 109 "U
;
2 1
F G 2 · 1010 "U
m\co :o:eb0 ! N "
1 F 71 G
≈ " F G
m\co :o:eb0 !**
1 F 71 G
F 1G 1028 ÷ 1034 "UF2 G
2 1
F9 1 G !-×!-
"
H "B, 86 / Y'
"A
> *% MJNOPQNR;
> 1 / 9 F3GO 1 C/ / 72 2 9 12 1 12 8;<;0 1 3 8;<;
I "B* . 1 1 1 '&'"!!UA!!
M 2 8 !
!
3 !
M!0 )* ÷ K, EcfWg 5 · 1011 3 · 1011W ≤ 60
3 . 14
! hg`^ X^Xac
- (5 N3&W 4 · 1013 ≈ 25
M V* N3&Wg 2 · 109W
i .
hg`^ X^Xac
Z 1 · 1012 K)),)
)* ' K N3&Wg 1
4
- 109W
K ' K) N3&Wg .
hg`^ X^Xac
$1 / 1 2 F])&f[Y0 '&' 'WG0 1/ C/0 1 / F /G / 8 1 / 1 / 921 C 5 / Q"!AR
6 1 1 1 238U 72 ∼ 2 37U F∼ 4 37 1 G ; 2 7 1
"B
Research Communities at FAIR
Nuclear Matter Physics with35-45 GeV/u HI beams, x1000
Hadron Physics
SIS 100/300
HESR
CBMHADES
Rare Isotope Production Target
100 m
Nuclear Structure & Astrophysicswith rare isotope beams, x10 000and excellent cooling
High EM Field (HI) _Fundamental Studies (HI & p)Applications (HI)
Plasma Physics: x600 higher target energy density 600kJ/g
Hadron Physicswith antiprotons of 0 - 15 GeV
Special Features:• 50ns Bunched beams• Electron cooling of secondary beams• SC magnets fast ramping• Parallel operation
SuperFRS
NESR
FLAIRCR-RESR
Target
AntiprotonProduction Target
H "B* 6 56 1 '& 1 O 5 ])&f[Y0 C / '&'",0 1 2 yW' 1 71 'W 1 6 7 ? / C/ F G '&'"!!UA!!0 1 F / 72 X'W0 YW0 / 2 )'WG0 /16 1 2 '\^b_yW' F|[)[0 Y(tG 71 $1])&f[YU'&'", 1 C '&'"!!UA!! 15 / 7 1
2 10 56 F 7 7 C 6 1 4 · 1010 UG ”H C” 1 !! B 072 / 1 N "!! 7U 37U[ 'W F7 /CG 1 1 C U 72 !, 37U
!! 2 1 2 62 12 2 1 y[&W 5 C H 8;<;
"-
I "B"! C9 1 / 1 /C 1 62 2 1 y[&W
# M 9 2 9.
. 3
T
K)YVP K)) (S N3&Wg U28+
FCFK)) !
G (5 N3& P)
. !!
( j * jW 5 · 10−12
.
K)YVP (V)) V* N3&Wg U92+ 2
FCFV))
>
cosθ hg`^ X^Xac
P j K jW . !!
,·10−12
.
#' (K( KV S*) 23&Wg -
=D BWkl(S+ V N3& p
G 240× 240
mm·m][h
∆p/p =
p ±3 · 10−2
!"
@ (*, KV S*) 23&Wg G
DnFD BWkl(S+ V N3& 1K TW4
@ p
=D
"E
I "B"" C9 1 / 1 /C 1 62 1 y[&W F1CG
@ ((( KV S*) 23&Wg
3 BWkl(S+ V N3& W
!0 e− *,) #3&
dnFD
+
0 p
!
3
3
@ ,S* ,) K* N3& 0
p K* N3& 3
3
<nFD p 5 N3&
S 4 ; $ T )T+;
; . V F|'&G 90 15 1/ / 720 1C / 72 F C 2 7 G > 1 C 5 1/1 1 2 1 F9 G1 1 11 F17 2 : C ” ”G 6 7 9 0 2 0 1
8 1 9 C 0 10 1 Q"!BRO / I"B"
UUU. 7 1C / 72 1C 1 2 7 1 19/ $
"+
I "B" C9 1 |'&
. @
>2 C F7G +
72 2 1 F7G -*-
1 / 2 1 r
2 1
F1 0 1G "-
2 1 9.4 · 1015U 2 ∼"
C 2 "*+-
L 56/ 7 / -!EA 2
2/ O / C 906 56 F 156 56/G <1F # 2 80 G
# 72 9 "37U 2/ / 5 0 1 2 / 1 1C / 72 1 F V0 2 1 ,!! 7 4 1 1 5 1 O %/ 4 9 [' C / C/ WX&YG
* RFVE )> <+;
> Q"!-R 1/C 2F; . V0 VG0 15 / 1 56 L 8 1 1 0 0 3He α 0 C 2 2 7
",
I "B"A C9 1 2 8
. @
L 2 E
@ C 15 E
1 "+- I
H C
H B
86 !!
L 56/ 7 / A!- 2
56 1C -!!
L 6 -! $ U
L A
1 ”yeqcc`^”
L ”yeqcc`^” *! N "-E 2
C 2 ∼ 60
”V” FσG 2 !A-
2 F /G B!!0 !!0 -"!0 B!! B!!·Q2U[ 7 ; / 1 N 1 µ[ N 2 1 C 1 C 1/ 1 0 3He 1 2 1 > 9 /9 11 1 71 1 0 1 1 C5 1 120 5 5 F 1G 8 2 10 156 71 ? / 1 1
"*
< 7 2 1 1 2 2 1 / /1 C/ F_qlm Wqdmbl 1 %9 1 N f['G0 2 /1 8 / 9 1 1 I"B"A0 "B"B 1 _qlm Wqdmbl C 1 2 6 10 1 1 6 1 / $C 1 1 1 1 /
I "B"B C9 1 1 2 1 _qlm WqdmblNf['
M E][oh D[phco ;BF
! V KS,
!"
M ! KP( S)
% ! −5 ' −90 −10 ' −135
2 0 B · ρ ,, T V( T
6 !! ,e V)e
N ! 0 )*( )*)
& ! 0 )*( )*)
2 !! VS)SP *5Y)
2 3 KW())))
1 KK N3&W 4
9." P)) K,)
C 72 10 WY)|0 SH
!
4,+2& /%38, 1+,C&,8,
FWXJNYJ )>7 T+;
> 0 Y'0 7 2 /C 1 2 / 1 / C / Q"!ER I"B"- # 5 9 6 2 0 2 O / F['[G 1 350 1 Y' 1C 71
I "B"- C9 1 Yf'&]'
M G
Tmaxkin 14 23& ,,) KVP)
∆p/p 2 · 10−4 2 · 10−3
8 14 1010 1011
? !0 .
!
1 × 4 (×K ,×(,
> * A * 1 -ZB>:A )-+;
"*-, 2 F .CG 9 2 1 2 0 92 4 I#H Ff)'G
"*+, 20 1 0 7 0 6 1 ;0 71 5 0 5 1/ F1 0 6LiG H I#H&&0 6 1/F 2 G 1 0 1 1 1 . "** 2 C
"
C 0 9 Z/A ≥ 0.22
I "B"E 1 72 1 I#H&&
U[ jdl Tkin/A0 jqv Tkin/A0
7U 7U
" "!! *-!
UA BE "+BE
"U E ""E*
!B "E+ ,A*
!A *B -A
! - E- A*
$ 1 1/ F I "B"E0"B"+G0 1 1 71 1 1/ 1 F 1 *!rG0 1/ F1 / 1 E!r0 N *!rG 1/ E F 5 1010 UG 1 2 "!! " 0 1 1 1 F1 72 1G B-
I#H&& 1 156/ 0 / 1 0 2 1 d → n > 9 1 2/ 56/ /
$1 /9 5 /10 6/ 2 1 1 95 7 f)' 1 C 1 1C / 72/ 2/ 9/ 7 9 1/ Y'F / 1 G0 1 9C 71 @ ! 1
1/ / / F Q"!+RG
$ C50 1 7 2 1 ? "**+ 20 ! 1 16I#H& I#H&&0 > 9 1 /0 1 / 1 6 / 1C / / 72 = F 2/ 1/ /0 H0 3 VG
I "B"+ ; 1 I#H&& F UG
L 1 .
p 7 · 1011 2 · 1011d 5 · 1011 3 · 1011
3He 4 · 10114He 4 · 101112C 109
14N 109
16O 2 · 10820Ne 2 · 10840Ar 108
84Kr 2 · 106 F30+G84Kr 8 · 106 F26+G128Xe 2.5 · 106
! " 'B%4 BCB
; 5 12 8;<; / 1C / 72 !"! !"+ 2 1 2 56/ 1?
• .61 / 2 2 1 0 ?
A
= 1 / 1/?
∗ $J 5 F” 5”G5 50
∗ 1/ 1 ”C ” F 0 1 75 56 $J 0 UG
= . 9 F G?6 5 D
• 7 / F 1 1 56 0 5 5O 21G
• . 2 9/ / 1 / / C/
• ; 1/ F6/ 1G / 29/ C F 2/ C/G
6 7 12 / 0 C9 -E 8;<; 1 1 / C/0 556 0 1 C/ ;$ 56 2
$ / /16 2 2 C 5 B- I 1 ≥ 340 O 1 1/ ∼ 2.0 I 6 / 1 1O 7 / / 1C t|F2 2 1 56 $J G '| F 1/ G
$1 ;$ C 1 ?
" 1 71 1 / / 72 1"÷BE 37U F C 5 pA GO
B
1 1 1 F1 1 G / 72 1 -÷" A 37 F C 5 p↑A GO
A 1 1 F C 5 d↑A G / 721 ÷-+ 37U
. / C 1 71 10 71 1C F G 90 9 /
H 1 1 72 F 1 /G? 1
C9 5 171 1 / C/ 1 [\0 | ] 72 "÷B- 37U 1 1 ·1027 −2·:−1 F1 Tkin ≈ 3.5 37UG 1 C 1 C
. 1 1/ 1 1 C 1 C 1·1030 −2·:−1 1 1 72 / 1 1 1 1/ 1 1 1 1/ / 1
1 1/ 1 71 16/ 9/ 2 1 1 1/ 0 1 5
/ 1 1 1 1 / 1 0 1 562 / 7 / 9 . 1 156/ / 1 1 0 ” ”
-
) '!"+&(
, !& "
8 1 1/ 1 1 1/ 1 1/ 0 Q,,R
. 1 1/ 1 6/ O0 10 1/ 1 7 1 2 /2
% r ? C0 1 1 0 10 6 rz(θ)02 θ 1C 0 0 7 1 2 2 OZ 2 θ C0 6 1 056 1 OZ
8 Ar 1 60 12 / A @ 0 C 1 K 0 0 1 0 2 K . K 1 92 0 7 C0 / 2 7 5 0 2 ” ” ”1” 1 1 $2 C K 92 C 2/0 7
E
1 1 F” ” ”1”G 5
$1 Ari 1 2 A
r 1 Ai /2 A 9
Ari = Rij ·Aj F"-"G
3× 3 R 7 Rij 0 0 r 1 1 Ai 0 // 9 F"-"G 2
Ar = R ·A . F"- G
10 r = ry(θ) F6 2 θ 2 Y G0
R [ry(θ)] =
⎛⎝ cos(θ) 0 sin(θ)
0 1 0−sin(θ) 0 cos(θ)
⎞⎠ F"-AG
$ 10 1 / 2/ / / e(i)0 1/ / / 1 /0 C0 e(j)0 1 er(j) 1 e(j)
F"-"G C 1 er(j) e(i) 0
er(j) = Rij · e(i) =(RT)ji· e(i) F"-BG
2 RT 1 R F10 6 R 2? RTR = RRT = 1G
.0 F"-"G 1 2 R0 1 F F"-BGG N 1 7 2 6
8 1 0 C 1 2 2 C 2 K 0 5 5 0156 F 1 6/0 1 5 0 1 2 2G
8 ”” ”$” + $$
+
. A " 0 1 2 S 5 O Aj I 0
A =∑j
Aj · e(j) . F"--G
. O′ N 5 0 156 Sr0 1 5 S 6 r 0 1 er(l) 2
A =∑l
(Al)Sr · er(l) . F"-EG
I 0 F"-BG0
(Ai)Sr = Rij
(r−1
) · Aj , F"-+G
2 1 9
[R(r)]−1
= R(r−1
). F"-,G
; 0 r0 1 ! 0 52 / Sr S0 R(r−1)0 F"-"G
1 1 s 1 1 2 1 9 / // ;0 (2s+ 1) / 1 η(m) 562 ?
η(s) =
⎛⎜⎜⎜⎜⎜⎜⎝
10..00
⎞⎟⎟⎟⎟⎟⎟⎠
, η(s−1) =
⎛⎜⎜⎜⎜⎜⎜⎝
01..00
⎞⎟⎟⎟⎟⎟⎟⎠
, ... , η(−s) =
⎛⎜⎜⎜⎜⎜⎜⎝
00..01
⎞⎟⎟⎟⎟⎟⎟⎠
, F"-*G
1 5 1 s 162 1
χ =∑m
χm · η(m) F"-"!G
L χm 5 ”1 ” 1 χ . 2 F"-+G01 7 2 1 1 (χm)Sr 2 1 χm 56 ?
(χi)Sr = Dsij(r
−1)(r−1
) · χj , F"-""G
,
2 Ds(r−1) 5 (2s+1)2 1 211 6 r F80 0 ? D†D = 1G. 2 F"-BG0 / 1 F” ”G?
η(m) = D(s)m′m · ηr(m′) F"-" G
> C ? 0 1 5 O S η(m)0 1 5 Or0 56 1 S 0 11 ηr(m′)0 / /
; ? 2 5 O S 1 s |sm >0 5 Or0 Sr0 1 S 6 r0 |sm >Sr 0 2
|sm >Sr= D(s)m′m(r−1) · |sm′ > . F"-"AG
81 0 1 2 /2 2 0 C 2 1F6G |sm > |sm >r0 F"-BG0 1 |sm >r 0
|sm >r= D(s)m′m(r) · |sm′ > , F"-"BG
C 1 2 2 0 7 1 /0 0 1 U(r)0 1 566 r 56 1 ?
|sm >r= U(r)|sm > . F"-"-G
8 5 0
D(s)m′m (r) =< sm′|U(r)|sm > F"-"EG
C F"-"AG 1
|sm >Sr= U(r−1
) |sm > . F"-"+G
A " 1/23"G . σj 0 1 5 1 1 1 sj0 2 N 5 C
*
2× 2 7 / 2 1 s 6 7 / (2s + 1)2 6 1 0 / Si0 2/ .0 C 2 C G /1 1 χ 62 C / 6 1 0 / 0 1 56 65 0 6 7 1 0 5 / 6 / 1 I 0 1 6 C 1
χ =
(cos (θ/2) e−1φ/2
sin (θ/2) e1φ/2
). F"-",G
1 1 Pχ0 1
Pχ ≡ 〈σχ〉 ≡ χ†σχ , F"-"*G
C 0
Pχ = (sinθcosφ, sinθsinφ, cosθ) , P2χ = 1 . F"- !G
Pχ 1 1 χ 2 1 5 / 7 AG / 1 1 χ 1
Pχ =〈s〉χs
, F"- "G
1 1 Pχ0 [2(2s+ 1)− 2] / 1 (2s+1)2 1χ
0 Si0 1 56 1 1 si 1 s 15 1 6/ 0
D(s)(r)SiD+(s)(r) = Rij
(r−1
)Si . F"- G
BG 1 1/2 9 F"- G /9 7 0 1 0 56 C FG Sr 56 S?
〈σi〉Sr = Rij
(r−1
) 〈σj〉 . F"- AG
A!
0 3 1
1 s 0 2 1 0 C 56 1/ 1 s 0 1 / C 1 0 .452 1 C 0 1 22 Ji0 2 1 6 /5 16
.7 1 C0 0 1 |ssz > 1 s2 sz 80 C6
1/ 15 12 1 0 1 1 56 4 1 H 20 0 7 1 9 7 1. C 1 C0 1 0 1 1 6
8 1 1 4 l 11C 6 r0 4 1 lj0j = x, y, z
0 1 1 C10 1 S 2 y 2θ F 15 S′G0 41 5 S′′0 2565 5 v Z S′0 1 1 S S′′ C1
S → S′′ = lz′ (v) ry (θ)S . F"- BG
L 4 1 1 1 l(v)0 20 1 2950
l (v) ≡ [r−1 (v)] ′′lz′ (v) r (v) . F"- -G
@ r (v) 6 2 e(z) × v 0 1
Z 1 v0 [r−1 (v)
] 0
1C ” ”
A"
B A 1 4 1 l0 1 7 Al$1 Al 1 A 2
Alµ = Λµν(l)Aν , F"- EG
2 4× 4 Λ l $2 7 1 60 ?
Λ(r) =
⎛⎜⎜⎝
1 0 0 000 R0
⎞⎟⎟⎠ , F"- +G
2 R N 0 1 F"-"G C l 4 1 F” ”G v z0
Λ [lz(v)] =
⎛⎜⎜⎝
coshu 0 0 sinhu0 1 0 00 0 1 0
sinhu 0 0 coshu
⎞⎟⎟⎠ , F"- ,G
2 coshu = 1/√1− v2 tanhu = v
I1 2 5 SA0 A m 1 / |s, sz > . O N 5 0 C
6 5 −v SA v = p/√p2 +m20
2 p F1G 1 I2 5 O0 26 A0 1 SA0 0 C65 1 p .7 0 1 0 1 7 p0 |p... > 8 0 5 O 2 S0 C 5 −v SA 7 2 2 C 1 S10 2 1 S1 6 S1 C 2 2 p <0 5 0 7 0 256 C 1 p0 / 1/ /0 / 1 2 2 I ”1”
' ( / %&$ ( %& $ TT
A
0 56 C A0 C 205 5 O > 1 2 29 6 5 0 ?
G ." 3 @ O 5 S 0 1 SA 4 1 l(−v) F"- -G 7 1 C0 0 |p, sz >
G ”0"” 3
. p 1 2 (θ, φ) I2 5 O 5 56 ? 1 S′0 C65 5
v = |p|/√p2 +m2
10 1 1C 15 Z AA@ 5 1 1 S′0
1 1 A 156 p = (p, θ, φ) . 9 6 7 2 N 2 (−θ) 2 OY ′0 2 (−φ) 2 Z N OZ ′′0 0 1 1
SA → S = rz′′(−φ)ry′ (−θ)l−1z (v)SA F"- *G
@ 0 1 r(α, β, γ) 6 7 2 α, β, γ0
r(α, β, γ) = rz′′ (γ)ry′(β)rz(α) , F"-A!G
S = r−1 (φ, θ, 0) l−1
z (v)SA . F"-A"G
A 1 SA |; s, sz = λ >0 2 =(m, 000)0 O0 1 S0 |; s, sz = λ >S0 |p;λ >0 |p;λ >
|p;λ >≡ |; s, sz = λ >S , F"-A G
S 1 9 C 1 F"-A G C 7 0 2 1 1C0 1 0 7 16 8
AA
" 0 2 1 15 0 7 10 15 1 1 1/ 4 1 1 F"- -G
B" "3# F"-A"0"-A G 1 1
2 1/ 0 1 S 2 0 A C0 56 |p;λ > I20 2 F"-A"G F"-A G0 7 5 SA0 S F"-A"G?
SA = lz′(v)r (φ, θ, 0)S , F"-AAG
16 1 1 sz = λ % 50 5 / C/0 A ”" " ”
. F"-AAG 5 1 01 1 1 h(p)?
h(p) ≡ lz′(v)r (φ, θ, 0) . F"-ABG
2 1 1 h(p)0 6 56 0 0 1 C0 0 7 0 C 0
h(p) = r (φ, θ, 0) lz(v) . F"-A-G
”9 /” 6 . 1 50 1 h(p)0 B 0 1
2 p I10 / /0
S = h−1(p)SA , F"-AEG
1 2 F"-"+G0
|p;λ >≡ |; s, sz = λ >S= U [h(p)]|; s, sz = λ > , F"-A+G
2 U 1 0 56 4 15h(p)
80 1 0 |p, λ > 1 F"-A+G0 1 C 15 1 ” ” λ
AB
C" " 3. C 71 / / 5 / 5 5 5 / / 0 0 .7 1 0 1 5 1 1 4 > / C 20 2 56 2 FI 10 2 6 1G
82 . S 5 O AC6 1 p 1 5 λ0 2 |p, λ >
. Sl N 0 1 4 1 l S . 1? 5 Ol 1 C AD
. 2 6 F F"-"+GG0 5 Ol0 1
|p, λ >Sl= U(l−1) |p, λ > , F"-A,G
2 U(l) N 1 0 956 1 4 l8 p′ 10 11 5
Ol A 2 1 p′µ 5 0 0 1 p0 / Ol F F"-+GG?
p′µ ≡ (pmu)Sl = Λµν
(l−1)pν . F"-A*G
80 C C 0 |p, λ >Sl= |p′, λ′ >02 p′ C F"-A*G0 1 7 λ′D L 7 10 1 1 1/ F"-A+G 19?
U(l−1) |p, λ >= U
(l−1)U [h(p)] |, λ > . F"-B!G
#C 7 2 U [h(p′)]U−1 [h(p′)] = 10 2h(p′) 1 10 C0 1 |p′, λ >?
|p′, λ >≡ U [h(p′)] |, λ > , F"-B"G
10 20 F"-B!G?
U(l−1) |p, λ >= U [h(p′)]R|, λ > , F"-B G
A-
2 R
R = U−1 [h(p′)]U(l−1)U [h(p)] ≡ U [h−1(p′)l−1h(p)
]. F"-BAG
; C F"-BAG 0 1 /10 ”1 /” U 0 0 90 20 l > 16 2 0 1 h−1(p′)l−1h(p) B = (m, 0, 0, 0, ) ?
qG h(p) : → p
G l−1 : p→ p′
:G h−1(p′) : p′ →
I 0 1 FqG0FG0F:G → 0 6 C .7 R 1 6 20 l 8 7 6 r(l,p)?
r(l,p) ≡ h−1(p′)l−1h(p) . F"-BBG
2 5 ”2 1 1 ” 1 l "0 162 p p′ F7 C 0 ”2 1 ”0 K CG
; F"-"BG F"-"-G 0 6 1 16 .7
R|, λ >= D(s)λ′λ [r (l,p)] |, λ′ > F"-B-G
1 Dλ′λ 0 1 7 F"-B G0 F"-A,G F"-B"G 1/ 95
|p, λ >Sl = D(s)λ′λ [r (l,p)]U [h (p′)] |, λ′ >=
= D(s)λ′λ [r (l,p)] |p′, λ′ > . F"-BEG
> 90 C1 C / Sl S 7 1 52 4 1 S Sl . 20 |p, λ >Sl |p′, λ′ > 6 0 * %7 ! S 0 ! SlT
AE
0 ”D! ”3/ 9 ; 71 0 2 l 1 6 Y 0 C/ C 1 0 7 r(l,p) ≡ 10 2 ”2 1 1 ” I 0 l = ry(β)0 p = (p, θ, 0) p′ = (p, θ − β, 0)0
|p, λ >→ |p′, λ > . F"-B+G
0 2 p 1 1 60 1 F"-B+G C
|p, λ >→ exp[−iαJ · p/|p|
]|p, λ >= e−iαλ|p, λ > . F"-B,G
$ % & 0; > 1 C . ? A + B → ... + K + ... ” ” FG 0 B 1 % C0 1 FG 12 2
H 0 1 0 m0 C5 1 2 φ = 00 1 XZ0 1 72 0 0
p = (p, θ, 0), E ; pL = (pL, θL, 0), EL . F"-B*G
; F"-A-G 2 ?
h (p) = ry(θ)lz(v) ; h (pL) = ry(θL)lz(vL) , F"--!G
2 v vL 8 Z 2 1 2 1 N
0 .7 1 S SL ” ” S 2 1 Z 5 βCM 0 2 βCM ; 0 S 1 l−1
z (βCM )0 1 SL. l F"-BBG0 l = l−1
z (βCM )0 p′ ≡ pL 1 6 r(l,p) ≡ r [l−1
z (βCM ),p]C /
1 N 2 ω F2 G; 0
r(ω) = h−1 (pL) lz(βCM )h (p) . F"--"G
A+
80 6 1 1 2 Y 1 1 7 1 92 920 1 1 F"--"G e(y) = (0010) e(x) = (0100) 1 Y X 80 1 e(y) .7 r(ω) = ry(ω)0 2 ω 16 2 0 1 10 92 e(x) = (0100) 1 1 F"--"G e(x) 1?
e(x)′ = 0, cos θ cos θL + γCM sin θ sin θL,
0,−mE
(sin θ cos θL − γCM cos θ sin θL).
0 1 1 e(x) 1 6 Y 2 ω0 7 2 C 9
cosω = cos θ cos θL + γCM sin θ sin θL F"-- G
sinω =m
E(sin θ cos θL − γCM cos θ sin θL)
γCM =1√
1− β2CM
.
I 0 1 1 2 ?
|p, λ >→D(s)λ′λ [ry(ω)] |pL, λ
′ >= dsλ′λ(ω)|pL, λ′ > . F"--AG
8 0 1 B 0 56 FTG 2 5 420 2 ω C 1 20 1 1 F Q,,R Q*!0 *"0 * RG 56/ C / /0 1 C C . 7 15 56 1 Fp EN 1 72 G?
tanh(u) =p
E; tanh(uL) =
pLEL
; tanh(uCM ) = βCM .
sinω
sinhuCM=
sin θLsinhu
=sin θ
sinhuL, F"--BG
8
A,
2 1 9 pL sin θL = p sin θ0
coshuCM = coshu coshuL − sinhu sinhuL cosω . F"---G
A * 0 ω < (θ − θL)? ω =(θ − θL)
6 0 2 K 1 1 p 1 2 φ / F G / C 0 7 2 1 1 1 ry(ω) φ
56 1 N A+ B → C +D 8 2 C θL 2 D θR I2 2ω C 0 1 56 1 ?
cosωC = cos θ cos θL + γCM sin θ sin θL =
=
(pBpC
)(EC
EB
)cos θL +
m2BE
2C −m2
CE2B
mBpCpLCEB
sinωC =mC
ECsin θ cos θL − γCM cos θ sin θL =
=mCβCMγCM
pLCsinθ =
(mC
mB
)(pBpC
)sin θL
cosωD = − cos θ cos θR + γCM sin θ sin θR = F"--EG
=
(pBpD
)(ED
EB
)cos θR +
m2BE
2D −m2
DE2B
mBpDpLDEB
sinωD =mDβCMγCM
pLDsinθ =
(mD
mB
)(pBpD
)sin θR
.
2 θ 2 C 122 0 2 mB = mD mA = mC 0
2 1 ?
ωD = θR , F"--+G
C / F10 ppG0 2mA = mB0 6 16?
ωC = θL . F"--,G
0 C 0 1 / 72/ 2 θC → 00 2 s→∞
A*
, π ;
H 1 1/ 1 1 1 I = 0 / 1
| I = 1 , Iz = 1 > = | π+ >
| I = 1 , Iz = 0 > = | π0 > F"--*G
| I = 1 , Iz = −1 > = | π− >
1 1 / 1 C 1 1 I = 0 , 1 , 2 7 1 1 1 ! ;11 C 2/ 1 "?
| 1 , 1 > =1√2
[ | π+ π0 > − | π0 π+ >] ≡| Π+ >
| 1 , 0 > =1√2
[ | π+ π− > − | π− π+ >] ≡| Π0 > F"-E!G
| 1 , − 1 > =1√2
[ | π0 π− > − | π− π0 >] ≡| Π− > ,
; 1 1?
| 0 , 0 >= 1√2
[ | π+ π− > + | π− π+ > − | π0 π0 >]. F"-E"G
I C | Π > : | π > 0 1 / 1 80 C
| 0 , 0 >= 1√2
[ | Π+ π− > + | Π− π+ > − | Π0 π0 >]. F"-E G
. F"-E G C F"-E!G0 1 F"" G C 1? 1 / 1
2 F"-E!GD8 ? 1/ 1/
C 1 1 1 0 , 1 , 2 /C C 10 0 10
!
B!
1 /1 1 1 10 1 1 11 2 10 1 0 C0 03π 1 1 10 2 3 I 1 1 C 0 1 C 1 0 /1 1 1 0
, . >
/ 1 0 1 N 156 / C 8 1 8 0 /> N ” 9” C 0 5 9 C 1/ 2/ 9 1 2/
6 1 10 5 56 1 ” 9” 8 12 C 1 0 1 ”1 9”
H 5 ” / 2/” Q*+R. 2 FA0 B0 C0 DG 1C 2/ I / C 2 9 C 0 1 C 1/ 52 2 2 5 2
@ 5 F6 2 π/2 π 1C 2 1 G0 F1 1 1 5G 9 C C 7 50 10 "-" CC21 0 0 4
√2 ≈ -EE > 90
9 "- 0 2 1 E $ 20 "- 1 0 ? 1/ 1 6 π/20 π
2 0 C/ 9 5 1 2 2 C 0 12 C? 1ϕ = π/4 0 1 "-"0 1 ϕ = 0 N "- C 1 0 "-A
B"
H "-" 9 "
H "- & 0
H "-A 86 16/ 1 2
9 F 1 2 7 G > 2 0 62
80 1 2
L(ϕ) =4
cosϕ+ 2 (1− tanϕ) . F"-EAG
B
7 2 / 1?
∂L
∂ϕ=
4 sinϕ− 2
(cosϕ)2 = 0 ; 4 sinϕ− 2 = 0 ; ϕ =
π
6, F"-EBG
0 C "-A0 9 1 1 ϕ = π/6 7 9 / T
0 9 C 2 1C 211 0 0 1 2 π
1 0 9 ? C C 1 2 ϕ = π/20 1 2 9 F 1 C 11C G I 0 6" / 5/ 0 9 O 9 C1 20 2 0 1 10 56/ > 1 ” ” N 2 9C 0 C / 9 211
0 / 90 56/ 5 20 2 1 0 / 9 9 ”C” 1/ 1 9 1 > 1 1 ”1 9” O 1 0 / 9 F8 / / 1 ”1 2” % 1 / / G
C 2 1C 2 9 1C 20 F G 1 10 5 ” V ” ” 9 ”
C 0 0 1 56 95 0 56 2 ?
" C / 0 C / 5 2 " !O
C2 2 / 0 0 1 / 0 2 C 9 " !0 " !O
BA
A 9 5 5 5O
B 1 9 2
. 7 2 1 % % Q*,R0 C / Q*+R
, ”B& ”D
9/ / F 2/0 / 2/G C 2 1 0 ” 2 ”0 1 ” 0 2 72 "!7” 1 8 0 1 1 C 10 4%8 QER
% ” ” C 1 562 5 ; 0 B 721 P = (E,p)0 B 1 0 1 72 N /0 7 1 1 1 / 6/ 1 1 ” ” B P P2 = m2? O 1 / 6/ F 1 15TG 0 6 ” ” 0 2 ” ”
C0 C 1 2 QAR0 1 F 0 1G C ” ” C C 0 C
BB
4 6
4 6 "
/ E"
2 1 1 6 0 0 9 / 2/ / 1 2 10 1 I C 19 2 1 2 0 2 0 C 0 N C 2/ 1/0 26/ 2 56/ 2 80 1C 9 / 6 25 1 C/ 1/ / 2/
/ 2/ 1 0 / 1 0 1C 1 2 ;/ 9 ? 0 / 8 / 2 C / 2 /0 0 2 7 5 1/
BE
# 0 1 C 1 1 F 1 0 2 GT .7 0 1 0 1 2 .0 N C C
2 0 12 1 / 2 1 1C 80 10 1 2/ 2 > 5 K 5 2 7 / 1 2110 0 1 . 9 9C / 1C 1 2 0 6 C 2 Q"0 0 A0 BR
/ B "
; a+ b→ c+ d C FAAG
s+ t+ u = m2a +m2
b +m2c +m2
d .
; 725 b s 01 0 25 FA +G
; . / 2 0 56 C 2 122 5 9 R F FAEGFA,G0 0 20 FA*GG?
dσ
dt∼ πR2
−t · J21
(R√−t) ,
?
" / dσ/dt | t | 1 72 0
/ dσ/dt 72 1 | t |0
A / dσ/dt 99 1 72 0
B+
B 1 1C5 dσ/dt 1 /R√| t |
; 0 565 0 1 2 θ∗ 0 C 0 1 | t | F 0 C dσ/dΩ∗ dσ/dtG H ?G F 1 1 G0 G F C0 10 1/ 0 2 1 / 11 1 G
; .1 1 1 u0 C 1 C0 1 C 1 t
"; H 12 a+b→ 1+2+3 0 1 a mproj =M1 9 b Mtarg FMX =M3+m2−Mtarg C ”C” G . FA"-G 12 72 FA"EG0 FA"+G0 FA",G 12 10 20 3 B1 C pp FG F5 1 pp→ p+ Λ+K+G0 3 FG MY F G ∑mi
; "G .C 0 C φ 16 1 F γ+p→ φ+p G 1C2 φ F 1C 1 φ? mp ≈ mφ ≡MG 12 1
3
4·M ≈ 750 7U .
G $ 2 12 2 1 C/φD . 2 1 95 151 D
#; @ + 1 6 1 C γ+p→ M+p 0 2 M M
B,
F10 7 G . 9 m 1 ? "G125 725 725 0 G 1 12
; . 16 1 72E 1/ 12 C 0 F 126GF$0 C/ 2 1 1 0 7 0 19 1 0 ”1”G $ 7 D
,; . /0 1/ 71 1 C5 φ p(γ, φ)X 1 1”/” 0 1 11 1 2 / 7 1 / 72/ 1 10 1 "E" 2 C 1 1 1 1 7 1 K . 7 1 1 1 φ → K+K− / 1 "E" 1 1 72 1 C0 7 1 C 0 1 1 7 1 7 CD L C 2 D
; . 1 9 FA G F 1 β 10 γ ≈ 1G?
u012 = (u1 · u2)u12 = u1 − u2 · u
01 + u0121 + u02
.
FV[? u01 = E1/m1 = γ10 u1 = p1/m1 = γ1β1 G
; . s 0 0 172 56/ 0
s = (Pa + Pb)2= (E∗
a + E∗b )
2,
B*
H "E" ? . 3 3 !0 !0 & 3 !0 ”0”
2 / 0 20 1 . 0C56 E∗
a E∗b s
; FqG . FA *G 1 a b FG . 25 1 F aG 1 b F G $ 1 1 FA *GD . 7 0 1 ” 5” 0 1 a0 1 b A3 ”= 2” λ (x, y, z) 1 λ (x, y, z) =
x2 + y2 + z2 − 2xy − 2xz − 2yz = (x− y − z)2 − 4yz
; FAA G0 C56 F 2 G 7 / 2 1 1 9/ F15 G s
-!
; a + b → a ′ + b ′ / Fma =mb = mG βcm F 0 2 b 1 G 5 a F bG β ∗
"; a + b → c + d 0 C562 1 c0 1 (px, py = 0, pz) FZ 1 1 1 1G C 71 F ”711”G .C 0 7 21 c 5 FAA,G0 2 β∗ c ?
p1 = (0, 0, γcmE∗ (βcm − β∗)) , p2 = (0, 0, γcmE
∗ (βcm + β∗)) .
; . n(K,π)Λ 56 1?
• 12 72 O
• K 1 pK F 0 9 12G C2 C2 1 Λ21D
• C 7 π ”” D
• C 7 Λ21 ”” D
• . 2 C2 1 Λ21 1 pK
#; @0
p⊥ = p∗⊥pz = γcmp
∗z + γcmβcmE
∗
E = γcmE∗ + γcmβcmp
∗z ,
C 2 c 2 C 0
tan θ =sin θ∗
γcm (cos θ∗ + g∗), g∗ =
βcmβ∗ .
-"
; 0 10 725 c 1 1 725 0 C 1 0 156 1 7 2 0 5 p(θ)?
E∗ + βcmγcmp cos θ = γcm(p2 +m2
)1/2.
H9 7 F FAB!GG
,; 12 1 µ+m→ µ+M 0 2 µ N 0 m N 90 M N 0 51 1 90 tmin0 0 s m2
i Fmi N 0 56/ G
/ ?" & , /
; FqG .C 0 l0m0 n0 1 F"E"G0 11 2 2 2 1 25 1 0 1/ 2 F"EAG
l =kf + ki∣∣kf + ki
∣∣ , m =kf − ki∣∣kf − ki
∣∣ , n =ki × kf∣∣ki × kf
∣∣ , F"E"G
2 ki kf 0 1 1 0 l0 m n01 F"E"G0 5 9
l = m× n , m = n× l , n = l×m . F"E G
6 C 2 1 /
l =kf + ki∣∣kf + ki
∣∣ , m =ki − kf∣∣ki − kf
∣∣ , n =ki × kf∣∣ki × kf
∣∣ . F"EAG
. 1 5 9
l = n×m , m = n× l , n = m× l . F"EBG
FG . F"E G F"EBG
-
F:G $ 7 / / 5 1 D 0 2 D
; 8K 0 1 5 2 D
; ”1 ” / 2 112 0 1 2 BAFG H 9 ”” ”1 + ” 1 1 FFB""GG 1 F 10 1 1 1 G 0 2 / ” 1/ ”0 1 9 E2
n − q2n = m2n
FG .C 0 ?P2n ≥ 0 2 FB" G
2 1?
q ≤ 3
4mN .
; 122 1 1 + 2 → 1 ′ + 2 ′ 2 2 2 19 0 565 2 7 / 2 F2 1 2 G
; H n(K,π)Λ "*E!/ 20 1 1 QE!R0 9 1 71 / 1 5 210 2 0 1 ;.2
;? 1 1 2 1 K0 9 Λ 16 7 ”2” 1 K . 15 65 /2 1 2 1 D
"; H 5 C 16 F MG?γ +N →M+N 81 0 1 72 Eγ F 1 9 C M G 7 C 16
-A
/ ?" & 3 5
; C 0 7 211 0 6/ 10 5 F3;$1G
#; C 0 1 C / 15 0 15 5 2 F3;$1G
; 1 0→ 1+ 2 0 M0 1 2 m1 m2 0 19 1/72 1 20 / / 720 C 97 / / 72
,; . ” ” 0 B1 P0 = (E0 ,p0) F 1 Z 115 1 p0G0 1 1 1 1 B1 P1 = (E1 ,p1) . 2 7 1 1 2 C p1 1 Z L 2 0 (P0 · P1) 1 156 ?
(P0 · P1) = E0E1 − p0p1 = E0E1 − p0p1 cos θ1 = E∗0E
∗1 .
. 72 1 156 0 72 1 1 0 9 FEBG C 0 56 1 1 2 .C 0 9 FEBG FE-G FEEG?
p1 =M0E
∗1p0 cos θ1 ± E0
√D1
E20 − p20 cos2 θ1
,
2D1 =M2
0 p∗12 −m2
1p20 sin
2 θ1 .
; 1 1→ 2+3 1 0 A 5 F1? 1 π0 G 9
sinψ23
2=
m
2√E2E3
-B
1C 2 1650 2 ψmin23 1
0 2 θ2 = θ3 F G L 7 2 2 2D F C ABG
; C 2 1 725O 7 C F AG 2 &&F12 G 2 3;$1 Q R
; . 2 ψ(E1) 1/ 1 0→ 1+2” ”0 2 B1 156 (E0 ,p0)0 1 2 C 1 p1
p2 1 72 1 1 E1. 0 16
FE" GFE"BG?
cosψ =E1E2 − q2
p1p2,
cosψ =E1 (E0 − E1)− q2√
E21 −m2
1
√(E0 − E1)
2 −m22
,
1 1 E10 0 5 1 1 0 ”” FE1 ,G ”1” FE1 ,G 0
E1 ,min/max =E0E
∗1 ∓ p0p∗1M0
, E1 ,min ≤ E1 ≤ E1 ,max .
; .C 1 0→ 1+2 ” ”0 0 21 N 0 FE"EG FE"+G?
E1 =E0
2±√E2
0
4− M2
0
2 (1− cosψ),
E2 =E0
2∓√E2
0
4− M2
0
2 (1− cosψ).
. 7 ? 2 9D$ 2 D
; 122 1 1 + 2 → 1 ′ + 2 ′ 2 2 2
--
19 0 565 2 7 / 2 F2 1 2 G
"; "G . a+ T → c+X 0 56 12 1 1 a 0 1 165 9 T L c 2 0 X C5 M F 565 G 2 9 C 9 2 01 1 2 2 6 1 0 ” ” 1 F 1 c 5G G 1 7 90 D
/, :
; . 0 565 1 F*"-G?∫
d4Pi δ(P2
i −m2i
)=
∫d3pidEi δ
(E2
i − p2i −m2
i
)=
∫d3pi2Ei
.
#; .C 0 FG 12 0 2
√s
2 0 K F 0 2
√s 12 5 sthreshG 0
1 F* BG?
R2(s) ∼√√
s− (m1 +m2) =√ε ,
2 ε ≡ √s−√sthresh ”19 12”OFG 1 F2 C1G0 1 F* -G?
Rur2 (s) =
π
2.
-E
; H FqG n + p → M + d FG n + p → M +n + p 12 C M / K 7 / FRa
2 Rb3G 1 19
56 120 1 1C 9 Ra
2/Rb3
,; H1 Λ C 1 Λ→ p+π−0 1 Λ→ n+ π0 9 / K 1 1 7
; 8K / 1 "E "EA0 / QEER?
" . 1 ” ”0 5 0 2”2” D
. 1 1 D
A 2 1 2115 ” ” 71D
B . 7 12C 0 5 D
- $ 1 2 5 5 p1 = p2D
E $ 0 1 1/0 10 1 1 1D
; . 2 .2 F +"G V 0 0 156 C/ 0 1 1 1 Λ0 5 / 0 2 1121D
// ?" &
; C 0 5 2 2 7 2 2 2
-+
H "E QEER 1 122 p p H1 1 1 1 F G 1 C 1C C O 1 C 1
H "EA QEER H1 1 1 1 2 / 1 1/ . C 1 C2 1
; / 2 1 P → 1 + 2 + 3 56 1? Ω3 = (cos θ13, ϕ3)0 15 5 1 3 p10 Ω1 N 51 1 1 .C 0 1 F"!EG?
d3p1 d3p3 = p21dp1 dΩ1 p
23dp3 dΩ3 = p1E1dE1dΩ1p3E3dE3d cos θ13dϕ3 ,
/ 2 2 K C 1 C F"!+G
R3(s) =1
8
∫dE1dE3dΩ1dϕ3 Θ
(1− cos2 θ13
).
@ Θ N 2 cos θ13 2 @ cos θ13 = ±1 5 2 1 (E1 , E3)0 2 2
; . F"!,G0 156 2 2?(√s− E1 − E3
)2= E2
1−m21+E
23−m2
3±[(E2
1 −m21
) (E2
3 −m23
)]1/2+m2
2 ,
-,
C 11 F"!*G
(√s− E1 − E3
)2=| p1 ± p3 |2 +m2
2 .
"; .C 0 2 2 F F"!"BGG
λ(p21 , p
22 , p
23
)= 0 .
; H 5 1 a+ b→ 1+2+3 2 1 "!*
7 2 1 1 ta20 tb20 ta30 tb10 s13 / 1 1Pi Pj # 1 1 2 1 F"!",G
/3 ?" &
#;$ 0 1 1 2 1 2 7 0 C 5 0 65 F/G F1G 1 2 1 60 7 F6 G 0 0 / 1/0 1 2 1 F""+G F"",G?
dρc.m.
dS= :`lkc
λ(p21 ,p
22 ,p
23
)4
,
1/ / C5
dρc.m.
dS= :`lkc
[1−
(1 +
2ε
(1− ε)2)r2 − 2ε
(2− ε)2 r3 cos(3ϕ)
],
2 ε 1 9 C
; . C F""""G 1 2 1 1 1 2
-*
1
dρc.m.
dS= :`lkc
[1− 2
2− ε r cos (3ϕ)
]r2 .
,; 19 5 12 2 12 2 1 0 1 1 1 N(0, 1)
; 86 1 " 2 " 912 0 2 0 1 1 N(X0, σ)
; 8 F" EG? 1 1 ρ = = 0 7 1 1 5 1 1 0 C / 7 1 0 1 . 1 1 σ 1 2 1?
" ρ0 2 7 1 1 H7
ρ
A ρ2
B 86 7 1 6 H7F" EG0 0 2 (x, y) 1 1 0 ? x 1 1 Nx(X0, σ)0 y N 1 Ny(Y0, σ)0 X0 = 00 Y0 = 0
; . 1 (X,Y ) 2 (x, y)0 1 x y 1 1 1 σ ! $ 17 / 1 5 r D $ r0 < r >
√< r2 >D
E!
;8 7 C 1 0 2 1 C 1 2 R
;. x y 1FCG 1 % 2 C 1 9 Γ?
w (z) ∼ Γ2/4
(z0 − z)2 + Γ2/4, F"E-G
2 z = x z = y 0 1 / 56/ F x0 = y0 = 0G
1 ” ” F 1 2 G 2 1 W (x, y) 7 / / 1 2 1 7 / 1 / / 1 ρ F10 !"0!A0 !-0 !,G0 12
ρi =W (x, y) = :`lkciW (x, y)max
.
C 2 1 0 1 $ 29 7 / / 1
/5 + "
"; B 20 1C/ 9/ C / C 2 0 52 2 C 1/ 5 20 17 1 C 2 C 7 F G 2 2 D
; ; 0 / / 35 5 / 1 12/ 0 9 / 0 25 11 7 ; 0 1 25 72 11 F 110 1 2 0 11 0 5 5 1C 5 1G
E"
H "EB & ! ! ! > / ! #! / 3 !
H "E- & ! ! M00 3 ! ! !
H ? 1 !, = ; C F2 u d 25 / G 0 F"G C 1 0 A C 1 0 1 0 F G C I 0 / 9/ 2 2
6 0 7 / 25/ 1 UA 2
H "EB"E- 5 C 0 C 25/ C 1 1 0 9 2 6 F1 7 2 656G
$ "EB"E- C 0 0 9 72 /2 D 8 #; . 1 / 1 1
115 1 6 x 1 15 1
" Pk(x) 20 1 1 k > 0
E
. 6 1 Pk(x) D L 7 D
A Pscatt(x) 20 1 1 /
; J 1 2 2 2 0 0
θmoliere =√213.6tbh
βcp· z ·
√x
X0·[1 + 0.038 · ln
(x
X0
)], F"EEG
2 p0 cβ z 1 F 7UG0 0 x X0 N 6 6 2
. 1 F"EEG 1 C
√2D
",; . 5 / 72 O 0 0 Z1 Z20 A1 A2 37U2
72 8 1C 2 2 10 562 1 C 1 ” ” F11 G 5 1C 2 2 1 2/ 56/ ?
" Au Au F GO
Cu Cu F GO
A d Au F GO
B 1 1
C 7 / 1 56/ // 1 72 C2 ?FG "! 37O FG " I7 8 / 1 2 1 7 / C
"; 8 F 3G 1 1 1 / 2 / 1/
EA
" ; 5 72 1 9 1 2 1 1C 5 1017 3 8 /0 / / 0 7"; 10 7 5 0 72 7 2 0 562 1 56/ 1 /C 9 C 0 2 C 6 8 /0 / / 0 7 8 F / =G 92 C 5 2 / 2 2
EB
/9 + ")
"; 10 1 /?
H "EE ? ./ Z ! 1 !0!4 K
E-
H "E+ ? ./ T !
EE
H "E, ? ./ M ! V @ 3 3 M 0 1 . ? 0 0
E+
/"&8 8
9
5 ; 1 / 1/0 ?
s = (Pa + Pb)2= m2
a +m2b + 2PaPb ,
t = (Pa − Pc)2= m2
a +m2c − 2PaPc ,
u = (Pa − Pd)2 = m2
a +m2d − 2PaPd ,
s+ t+ u =c∑
i=a
m2I + 2P (Pb − Pc − Pd) + 2m2
a ;
1 / 12 B1 Pa +Pb = Pc +Pd Pa = −Pb + Pc + Pd0 1 1 7 0 P2
a = m2a0
1/ FAAG
5 ; 5 9 R0 1
dσ
dt∼ πR2
−t · J21
(R√−t) ,
E,
• 72 20 O
• 1 t = 00 1 / t /0 R
√−t = Zi0 2 Zi i− % J1(x)0 1 ti = −(Zi/R)
2 C 0 / | t |O• 12 F dσ/dt(0)GG 1 1 R 9O
• ”. ” 2 F12 2 G0 1 d (ln (dσ/dt)) /dt(t =0)0 C 1 0 1 R 9 72 F G
L 1 1C5 dσ/dt 1 /R√| t | F C 1 1/C ”
”G0 / 1 1 %; . ?
Jm(z) =(z2
)m ∞∑k=0
(−1)kk!Γ(m+ k + 1)
(z2
)2k=
=(z2
)m ∞∑k=0
(−z2/4)kk!
· 1
Γ(m+ k + 1).
10 ex =∑∞
k=0 xk/k! 2 0
J1(2x) = x
∞∑k=0
(−x2)kk!
· 1
(k + 1)!;
J1(2x)
x− e−x2/2 =
=
∞∑k=0
(−x2)kk!
·[
1
(k + 1)!− 1
2k
];
J1(2x)
x= e−x2/2 − x4
24+O
(x6). F"+"G
E*
; .C J0 J1 C 165 ?
J0(xn) = 0 ⇒ xnπ
= n− 1
4+
0.050661
4n− 1− 0.053041
(4n− 1)3+O
([4n− 1]−5
),
J1(Zn) = 0 ⇒ Zn
π= n+
1
4− 0.151982
4n+ 1+
0.015399
(4n+ 1)3−O ([4n+ 1]−5
).
. J1(x) 1 C?
Z0 = 0 ; Z1 2.233π ; Z2 3.238π ; Z3 4.241π
6 FA,G F 2 9 x ⇒ kϑR = R
√−tG 1 C 1C 122 122 1 ?
x1 = kϑ1R 2.233π ; (kϑ1)2 4.986π2
R2 309
σtot;
| t |1 (kϑ1)2 309
σtotF"+ G
$ %' | t |; ; FA*G F"+"G C 1 5 1 5 / 122 0 1/C C +! 2 1O 10 1 7 1 7 1 5 % F Q+-RG "*-E 2?
dσ
dΩ=| f |2= k2R4
4· e−k2R2ϑ2/4 F"+AG
dσ
dt=
π
k2· dσdΩ
= π · R4
4· e−R2·|t|/4 =
σ2tot
16π· e−R2·|t|/4 F"+BG
. C 1 F"+BG 0 1 10 Q+ER 3 39 ; "*-+ 2 1 71 / / ; /1 61 7 1 / 122 1 / 72/ 1 ”1 2 ”
+!
b 5 9 7 0
b =R2
4=σtot8π
F"+-G
; H 12 ”” 0 ϑ∗ ",!
5 "; \+
T threshprojectile =MX+
MX
Mtarg·(mproj. +
MX
2
), MX =Mproj+m2−Mtarg .
. 15 12 ?
sthresh = (Pproj + Ptarg)2= (Mproj +Mtarg +MX)
2, F"+EG
2 MX ”C ”0 1 5 / ; 1 F 12G 72 2 sthresh F9 1 G0 C 12B1 2 ?
sthresh =M2proj +M2
targ + 2 (Tthresh +Mproj)Mtarg ,
2 C 0 Pproj = ((T +Mproj) ,pproj) Ptarg =(Mtarg,ptarg = 0) H 1 F"+EG 1 1 C C5 sthresh 0 1 C/ 2/ 1 1 5
]+ tp→meson 12 1 pp →p+ Y +jbk`l F FA"EGFA",GG
. 150 tp→meson = (Pa − Pmeson)20 2 a 1
F 1 Mp G0 Y FC 50 7 6 G MY 0 F7 G m
. I2 B1 F 12TG Pmeson =(m, 0)0
tp→meson = (Pa − Pmeson)2=(E∗ 2
a −m, p∗a
)2=
= M2p +m2 − 2E∗
a ·m , F"++G
+"
2 E∗a 72 12 1 0 pa 2
1 F C 0 G . 5 1 0 172 0 0 2E∗
a 8 C0 2 0 2 15 1s0
√s0 0 E∗
a =√s/2 .
120
E∗a =
1
2
√s =
1
2(Mp +MY +m) .
. 1 C F"++G0 1 9 2 1/ ?
tp→meson =M2p ·[1− m
Mp
(1 +
MY
Mp
)].
2 0 B10 1/ 1 ”1 1 N 1 ” ”1 1 N Y ”
5 ; # 1 1 ? "G 1 1 12 725 Ethresh
γ 12 O G βcm 1 Ethresh
γ O AG 725 1 C2 φ 0 1 10 1 1 1 φ . 1 121 0 1 1 21 5
5 #; C0 1 9 +0 0
M = m0 0 12 72
Ethrγ =M ·
(1 +
M
2m
).
C 1 9
βcm =p tot
Etot, γcm =
Etot
Mtot,
2 Etot = Eγ + m 1 720 1 1/ 0 Mtot 1 FZ9G0 0 1
√s0 p tot = pγ + ptarg = pγ
+
1 1 7 @ 725 12 1 1 12 0 0 1
|pM , thrlab | =M · 1 +
M2m
1 + mM
.
. 0 M = m 7 1 +
5 ; #? 1 0 1 1 9 ,
5 ,; #?
" 1 9 + * 1 0 / 1 "E" 0 1 0 / 16 O
/ 1 1 72 Eγ < 2 37 10 C 62 / 72 122C 1 2→ 2
5 ; 19 9 FA G 2 β 10 γ ≈ 1?
u012 = (u1 · u2)u12 = u1 − u2 · u
01 + u0121 + u02
.
7 u01 ≈ 1 u02 ≈ 10 FA G H 0 0 1 u012 = (u1, u2) = u01 · u02 − (u1 · u2) = 1 −(β1 · β2) ≈ 10 2 1 5 / ?
u12 = u1 − u2 .
5 ; 1 F2 FA *GG 19 C s ?
s = (Pa + Pb)2= m2
a +m2b + 2Eamb ;
+A
0
Ea =s−m2
a −m2b
2mb.
5 1 72? p2 = E2 − m2O 1 7 9 C5 s 725 1 1 2 0 1?
pa =λ1/2
(s,m2
a,m2b
)2mb
,
5 ; FAA G?
βcm =pa
Ea +mb≈ 1− 2m2
b
s, γcm =
s−m2a +m2
b
2mb√s
≈√s
2mb.
- ; 4 2 0 1 150 9 1 72 F G 7 F 7 1 G3 F G 0
γcm =Ea +mb√
s,
1 72 2 Ea + mb0 7 2 0
√s
C Ea s 1 16 ;1 0 1?
γcm =s−m2
a +m2b
2mb√s
=s(1− m2
a−m2b
s
)2mb√s
,
1 s m2a0 s m2
b 0 2 C 1 /0 C
γcm =s−m2
a +m2b
2mb√s
≈√s
2mb.
@ 0
γ =1√
1− β2, β2 = 1− 1
γ2,
+B
C 1 mb/√s 10 2 1
5 βcm?
βcm =pa
Ea +mb≈ 1− 2m2
b
s.
- ; . 15 2 β0 0 9 12 1 F G 1 72 . 72 2 Ea + mb 56 C C 1 9. 1 2 pa C 2 15 s 1 16 I 0
βcm =λ1/2
(s, m2
a, m2b
)s−m2
a +m2b
, F"+,G
0 s m2a0 s m2
b 7 2 15 λ0 19 ?
λ1/2(s, m2
a, m2b
)= s
(1− 2
m2a +m2
b
s+
(m2
a −m2b
)2s2
)1/2
,
C 0 C 12 1 1 1/s?
λ1/2(s, m2
a, m2b
) ≈ s(1− m2a +m2
b
s
).
2 1 1 F"+,G?
1
s−m2a +m2
b
=1
s(1− m2
a−m2b
s
) ≈ 1
s
(1 +
m2a −m2
b
s
).
C 0 1 C?
βcm =pa
Ea +mb≈ 1− 2m2
b
s.
C γcm0 1 5 C2 5 β
+-
5 ; ; "A C 0
pa =λ1/2
(s,m2
a,m2b
)2mb
, Ea =s−m2
a −m2b
2mb, Ea+mb =
s−m2a +m2
b
2mb.
. 7 C F 165 G C 0 ma = mb = m?
βcm =pa
Ea +mb=λ1/2
(s,m2,m2
)s
.
a 0
β ∗a =
p ∗
E ∗ ,
1 C 1 720 1 F FA +G FA *GG0 ma = mb = m?
β ∗a =
p ∗
E ∗ =λ1/2
(s,m2,m2
)s
,
?
βcm = β ∗a g ∗
a =βcmβ ∗a
= 1 .
; 0 ' 0%7 * 6 * 0 % )! ' + 3'
5 "; C 5 −βcm F ”” 0 1 1 1C 15 G I2 1 0 2 1 10
pz = γcm (βcmE∗ + p∗z) , p⊥ = 0 .
2 | pz | 12 1 0 | p z |=| β ∗ | ·E∗ 8 5
pz = γcmE∗ (βcm + β∗) , p⊥ = 0 ,
+E
1 C 1 1 C 0
pz = γcmE∗ (βcm − β∗) , p⊥ = 0 ,
C 1 1C 1
5 #; 80 2 θ C 9
tan θ =| p⊥|pz≡ p⊥pz
.
2 0 0 p⊥ = p ∗⊥ = p ∗ sin θ∗ p ∗
z = p ∗ cos θ ∗;1 9 16 0 2 0
pz = γcmE∗ (βcm + β ∗ cos θ ∗) , p ∗ = β ∗E∗ .
C 2 2 0 1 C?
tan θ =sin θ ∗
γcm (g ∗ + cos θ ∗), g ∗ =
βcmβ ∗ .
9 FAB!G
E∗ + βcmγcmp cos θ = γcm(p2 +m2
)1/2.
. 1 10 0 7 2 E∗ mγ∗0 1?
p2(1− β2
cm cos2 θ)− 2pmβcm
γ∗
γcmcos θ +m2
(1− γ∗ 2
γ2cm
)= 0 .
H 4D 7 2 ?
4D = 4m2β2cm
γ∗ 2
γ2cmcos2 θ − 4
(1− β2
cm cos2 θ) ·m2
(1− γ∗ 2
γ2cm
),
1 20 1 1 > ?
D = m2
[β2cm cos2 θ +
γ∗ 2
γ2cm− 1
].
++
I1 1/γ2cm / 0 C 1 9
cos2 θ + sin2 θ = 1 , γ∗ 2 − 1 = γ∗ 2β∗ 2 .
20 1 9/ 10 1 C
4D = 4m2
γ2cm
[γ∗ 2β∗ 2 − γ2cmβ2
cm sin2 θ]. F"+*G
I1 1 9 /2 F FAB"GG
p± =m
γcm· βcmγ
∗ cos θ ± [γ∗ 2β∗ 2 − γ2cmβ2cm sin2 θ
]1/21− β2
cm cos2 θ.
,; H 5 125 5 1µ+m→ µ+M 0 2 µ N 0 m N 90M N 0 5 1 1 9^ * %7 9 0 µ7 m M 7 3 6 6 ;I 0 B1 1 µ Pµ0 B1 7 C P ′
µ
. 15 1 t ?
t = (Pµ − P ′µ)
2= (Pm − PM )
2= 2µ2 − 2PµP ′
µ ,
C 1
t = 2µ2 − 2(E∗
µE′∗µ − p∗µp ′∗
µ cos θ∗) .
@ θ∗ 2 > 1 cos θ∗ ?
cos θ∗ =t− 2µ2 + 2E∗
µE′∗µ
2p∗µp ′∗µ
,
| cos θ∗ |≤ 1 2 1 / 1/ F 1 2
+,
2G . cos θ∗ +1 −1O 5 1 1 t 80 | tmin | cos θ∗ = +1 F 7 122 0 2 | tmin |= 0G
.C cos θ∗ = +10 1 tmin?
tmin = 2µ2 − 2E∗µE
′∗µ + 2p∗µp
′∗µ . F"+"!G
0 12 B1 t 0 12/ 0 1 112 1 F 7 122 G0 1 12 1 1/ 72 1 5 tmin 1 1 / 1 12 1 56 1 72
I1 72 1 C1 / C F L &&G?
tmin = 2µ2 − 1
2s
[(s+ µ2 −m2
) (s+ µ2 −M2
)−− λ1/2
(s, µ2,m2
)λ1/2
(s, µ2,M2
)]. F"+""G
1 C 2 1 1 1 8 1s m2
i 8 0 7 2 56 1 ?
εµm =µ
2Eµ=
mµ
s− µ2 −m2, εµM =
µ
2E ′µ=
Mµ
s− µ2 −M2.
? λ1/2(s, µ2,m2
) . 1
λ0 2 0
λ1/2(s, µ2,m2
)=
[(s− µ2 −m2
)2 − 4m2µ2]1/2
=
=(s− µ2 −m2
) (1− 4ε2µm
)1/2=
=(s− µ2 −m2
) (1− 2ε2µm − 2ε4µm +O
(ε6µm
));
2 C 1 λ1/2(s, µ2,M2
) 8
1 1 C C tmin 1 1 1 0 20
+*
80 C 1 F"+"!G . 7 12 C
(1 + x)1/2 = 1 +1
2x− 1
8x2 + ... . F"+" G
0
E ′∗µ =
s+ µ2 −M2
2√s
= E∗µ −
M2 −m2
2√s
, F"+"AG
17
E∗µE
′∗µ = E∗ 2
µ − E∗µ
M2 −m2
2√s
. F"+"BG
I1 1 p ∗µp
′ ∗µ . p ′ ∗ 2
µ = E ′ ∗ 2µ −µ20
1 5 E ′∗µ F"+"AG 0 1 9
?
p ′ ∗µ = p ∗
µ
[1− E∗
µ
p ∗ 2µ
· M2 −m2
√s
+1
p ∗ 2µ
(M2 −m2
)24s
]1/2.
I 0 1 p ∗µp
′ ∗µ
p ∗µp
′ ∗µ = p ∗ 2
µ
[1− E∗
µ
p ∗ 2µ
· M2 −m2
√s
+1
p ∗ 2µ
(M2 −m2
)24s
]1/2.
1 F"+" G?
p ∗µp
′ ∗µ = p ∗ 2
µ
[1− 1
2
E∗µ
p ∗ 2µ
· M2 −m2
√s
+1
2p ∗ 2µ
(M2 −m2
)24s
−
− 1
8
p ∗ 4µ + µ2
p ∗ 2µ
·(M2 −m2
)2s
+
+1
4
E∗µ
p ∗ 4µ
·(M2 −m2
)34s√s
− ...
]. F"+"-G
H E∗µE
′∗µ − p ∗
µp′ ∗µ 10 1 F"+"BG
F"+"-G0 C p ∗µ = λ1/2
(s, µ2,m2
)/2√s?
E∗µE
′∗µ − p ∗
µp′ ∗µ = µ2 +
1
8
µ2(M2 −m2
)2sp ∗ 2
m u+ ... =
= µ2 +1
2
µ2(M2 −m2
)2λ (s, µ2,m2)
+ ... . F"+"EG
,!
. E∗µE
′∗µ − p ∗
µp′ ∗µ F"+"EG F"+"!G0 1
s m2i 0 2 λ C C 10 1/
?
tmin ≈ −µ2(M2 −m2
)2s2
+ ... . F"+"+G
;FG . 1
2 F"E"G?
l =kf + ki∣∣kf + ki
∣∣ , m =kf − ki∣∣kf − ki
∣∣ , n =ki × kf∣∣ki × kf
∣∣ ,2 ki kf 0 1 1 0 F 0 G
/ 0 l0 m n 5 9?
l = m× n , m = n× l , n = l×m .
. 1 20 l = m× n
m× n =kf − ki∣∣kf − ki
∣∣ × ki × kf∣∣ki × kf
∣∣ = kf × [ki × kf ]− ki × [ki × kf ]∣∣kf − ki
∣∣ · ∣∣ki × kf
∣∣ .
H O 1 1 a × [b× c] =b (ac) − c (ab)?
kf × [ki × kf ] = ki (kf kf )− kf (kf ki) = ki − kf cos θ ,
ki × [ki × kf ] = ki (ki kf )− kf (ki ki) = ki cos θ − kf ,
kf × [ki × kf ] − ki × [ki × kf ] = (ki + kf ) (1− cos θ) ,
2 θ 2 C kf kiI1 ?
∣∣kf − ki
∣∣ · ∣∣ki × kf
∣∣ = sin θ√
2 (1− cos θ) = 2 cosθ
2(1− cos θ) ,
0
m× n =ki + kf
2 cos θ/2.
,"
80∣∣kf + ki
∣∣ =√2 (1 + cos θ) = 2 cos θ/20 0 0
m× n =ki + kf∣∣kf + ki
∣∣ = l .
2 0 / 0 5 1 9/ 9
; 1C 0 1 C FB""G
Preln =
(Md
mpEp −mp , − q
Md
mp
).
; 2 0 Preln = m2
n0 1 5 > 0 2 Z1 C 2 F G 1 1 F C G
@ 9 q P2n = 0 0
2 1/ 0 P2
d = (Pp + Pn)2 =M2
d T
; 19 / 72 /2 1 # 0 Λ21 1 F” ”G? 7 1C 9 1 72 K?
EK =m2
K −m2π
2(mΛ −mn)+mΛ −mn
2.
. 0 1 22 1 F -A! 7UG
"; C0 1 9 16 0 1
Eγ =m · (2M −m)
2(M −m).
; 72 0 ” ” ” .22” C 1 m < M 0 0 C2 C 1/ 12 9 9
,
; ? 19 1 7 211 0
#; ? 1 6 9 16
; . 3;$1 Q R F2 &&0 12 B0 1 F *GG
0; @ 0 ?
(P1 + P2)2= P2
0 ,
F E""G0 C 1?
q2 =1
2
(M2
0 −m21 −m2
2
).
; FE""G 2 0 1 0 FE" G 1 I1 E2 p2 E1 0 01 3;$1 Q R F2 &&0 12 BG
; . 1 1 1 7 0 1 ”6” C 50 2 ”2”0 C 11 K ”25”
1 9 1 1/ 0 16 2
#0; . 3 L Q,R
"; > F 12 "-A0 Q*+R0 N ” V ”0 ” 9 ” / C 1 ”$ 9 ” FC ”$ ”G H 1/ 1 0 / 9 0 C 1
;. 7 1 1 "EB8 0 / V
,A
$J 9 F 6 7 G 2 1 C5
; 8 7 "AE7 L 1 10 0 10 1 / Q *R
,B
:"
+ "!;"
(
"
: % ! 3%'
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,+
: % !
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Q*R 44 490 ”I 1” ?44 490 ”I ”0 &&””0 3 0 "*E+
Q"!R %% 0 490 4.. 0”H ”0 & ?44 490 ”I ”0 &h””0 3 0 "*E,
Q""R 30 ”= 7 / ”0 1 2 /0 0 ””0 3 0 "*E*
Q" R H % 0 ”= ”0 1 2 1 ;V10 ””0 0 "*+E
Q"AR C%0 0 ”H ”0 " 0 1 2 %8$ F "G0;2 F G 1 %% 20 0””0 3 0"*+,
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Q"-R J.0 ”= / ”0 1 2;;; 1 ;0 0 ””0 "*,A
Q"ER 4% 80 ”= 7 / ”0 0 ””03 0 "*,B
Q"+R =J0 0 ”$ 1 ”0 1 2 .$0 3320 4I; 1 M 0 0 ””0 "*,+
Q",R =$0 =%0 ” 1 ”0 1 2 %0 0 >2 0 "*,+
,,
Q"*R I>0 0 ”. ”0 1 2$0 $. 1 ;V100 ””0 3 0"**"
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Q "R ;I0 ” 1 / ”0 3#0 "**+
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Q AR .0 V0 ” 5 51”0 1 2 % 1 %0 ”H2 / ”0;C0 !!" 2
Q BR 20 ”$ 1”0 " 0 1 2O 1 L20 0 ”= ”0 !!A !!B
Q -R ; $1 0 ” ”0 A0? $$20 !!E
: %
Q ER $0 ”@ 1 7 / ”0 1 2 $..0 / 1 SV0””0 3 0 0 "*E,
Q +R 433 0 ” 1 ”0”9 9”0 0 "*+
Q ,R C$0 320 I20 ” 1 9”0 1 2 3 1 .$120 0 ” ”0 0 "*+-
,*
"+&
$ 10 1 ; 0 2 9 1 2 1 F6 56 25 0 1 5 1C 1G.1 0 6 5 1 / C / F N 7 /G ;5 5 0 1 1 / 7 / 11 Fzcc^?UUvvveqlex`U zcc^?UU_\q_vd`_xUG
*!
/!"8 +8 &
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( * 0
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QA"R 10 ” C6/ ”0 #=0F"*EBG0 fsss&h0 1"0 ","",,
( * 0"
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QA-R |[td_q:0 Wbt`m|zok F"*B*G ^A*
*"
QAER 'bdlb_x0 |zokWb , F"*EEG ^"A"A
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QA*R [|`\kzdl qlm h|'zbebkc0 |q_c qlm )\:e F"*,AG ^""BE
QB!R )|`ed`\0 [ll |zok F"*,*G ^AE+
QB"R =40 >L<0 " 1 - F"**!G " -"
QB R 3;40 >L<0 B 1 " F"**AG "B!
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*
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( * 0#
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( * 0
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(
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97 ; ;
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